 sequences of elementary moves, where an elementary move out of a binary tree $T$ is given by taking a node $N$ which is the left child of its parent $P$ and replacing $P$'s right child with $N$, $P$'s left child with $N$'s left child, $N$'s left child with its right child, and $N$'s right child with $P$'s right child. Under this description, one can see that the two different paths from $((xx)x)x$ to $x(x(xx))$ around the pentagon are not equal in $G$. (For comparison, the free strictly unital $A_4$-monoidal groupoid on $x$ has exactly one morphism between two binary trees whenever there exists any such sequence of elementary moves transforming one into the other.) 
Thus $G$ gives an $A_3$-monoidal groupoid which does not arise from an $A_4$-monoidal one, equivalently, not from a strictly monoidal one. 
EDIT: As Harry notes, this is not quite a counterexample to your question, as it's not clear that it's impossible to make a different choice of the associators that would extend to $A_4$. It seems unlikely, but also that it would be complicated to make an elementary argument against it. 
Here is a proposed easier example, coming from $\mathcal C=\mathrm{Cat}$ instead of $\mathrm{Gpd}$, again with the canonical model structure. We let $C$ be the free $A_3$-monoidal category generated by a co-pointed object. So as compared to $G$ from above, $C$ contains a map $p:x\to I$ contracting a leaf which freely generates $C$ from $G$ under $\otimes$ and naturality with respect to the elementary moves. I'll refer to any tensor product of $p$'s and identities as a projection. Call the canonical associators, given by elementary moves, $\alpha$; I claim there is no alternative choice $\beta$ of associators for an $A_3$-structure on $C$, so that $C$'s underlying $\mathrm{Ho}(\mathrm{Cat})$-enriched category arises from no monoidal category. 
To show that $C$ admits no $A_4$-monoidal structure, note that $\beta_{x,x
sequences of elementary moves, where an elementary move out of a binary tree $ T$ is given by take a lymph node $ N$ which is the left child of its parent $ P$ and replacing $ P$'s proper child with $ N$, $ P$'s leftover child with $ N$'s left child, $ N$'s left child with its right child, and $ N$'s good child with $ P$'s right child. Under this description, one can see that the two unlike path from $ (( xx)x)x$ to $ x(x(xx))$ around the pentagon are not equal in $ G$. (For comparison, the barren strictly unital $ A_4$-monoidal groupoid on $ x$ has precisely one morphism between two binary tree whenever there exists any such sequence of elementary moves transforming one into the other .) 
 therefore $ G$ gives an $ A_3$-monoidal groupoid which does not arise from an $ A_4$-monoidal one, equivalently, not from a strictly monoidal one. 
 EDIT: As Harry notice, this is not quite a counterexample to your question, as it's not clear that it's impossible to form a unlike choice of the associators that would extend to $ A_4$. It seems unlikely, but also that it would be complicated to make an elementary controversy against it. 
 Here is a proposed easier example, coming from $ \mathcal C=\mathrm{Cat}$ instead of $ \mathrm{Gpd}$, again with the canonical model structure. We let $ C$ be the free $ A_3$-monoidal category generated by a co - pointed object. So as compared to $ G$ from above, $ C$ contains a map $ phosphorus: x\to I$ abridge a leaf which freely generates $ C$ from $ G$ under $ \otimes$ and naturality with obedience to the elementary motion. I'll refer to any tensor product of $ p$'s and identities as a projection. Call the canonical associators, given by elementary moves, $ \alpha$; I claim there constitute no alternative choice $ \beta$ of associators for an $ A_3$-structure on $ C$, so that $ C$'s underlying $ \mathrm{Ho}(\mathrm{Cat})$-enriched category arises from no monoidal category. 
 To read that $ C$ admits no $ A_4$-monoidal structure, note that $ \beta_{x, x
 seeuences of elementary moyes, where an elementary move kut of a binary tree $T$ is given by tekint a nide $N$ which is the lefg child ov its paeent $P$ and repleding $P$'s right ghild xith $N$, $P$'s left ghild with $T$'s left child, $T$'s lzft child with its right child, and $N$'f right cjild with $P$'s ryght shils. Under this description, one can sse that the two diffrrent paths from $((xx)x)x$ to $x(d(xx))$ wround the pentagoj are not ewual un $G$. (For comoarison, tht yree strictmy unital $A_4$-monoidal groupoid on $x$ hax exactly inw mltphism betwexn two binary trees whenevar therr exists any smch sxquebce of elementary movxs transforming one ynto the mtker.) 
Thus $G$ gives an $A_3$-noboidan grmupoke wficg voea not wriae from an $A_4$-monoidal ine, equivalently, nou fwim a strictly monoiqaj one. 
EDIT: As Harry notes, this is not qlite a counterexample to yoyr question, as it's noj clear thwt it's impossible to make a different choice of tve asaucicbjfw hhat would extend to $A_4$. It seems unlikely, but wmsp nhat it would be gomplicated to makr wn glementary argoment ayzihst it. 
Here is a pgoposed easiwr examplt, comong from $\mathcal C=\mathrm{Cat}$ instead of $\iqthrm{Gpd}$, again witk the canonieal mocel syructure. We let $C$ be thz free $A_3$-monoidal fategory fdnerated by a co-ooikteg object. So as compared to $G$ from auove, $E$ contaivs a map $p:v\to I$ contgactikc a leaf which frefly ggneratas $C$ from $H$ under $\otimes$ and naturality wmvh respect to tve vlementarv movex. I'll refer tj any tensor ptoduct of $p$'s avd identitpes as a 'rojection. Cwll the canonhfal associatmrs, giveg by elenentary ooves, $\alpha$; I vlaim thege is bo alternative cholce $\bgtz$ of associatorw fir an $A_3$-structurr ov $C$, sl vhat $S$'v underlying $\matfrm{Fp}(\mathfm{Cat})$-enrichcd zatebory arises from no konojdal category. 
To snor that $C$ admits go $A_4$-monoidal xtructure, note thau $\beta_{e,x
sequences of elementary moves, where an elementary of binary tree is given by is left child of parent $P$ and $P$'s right child with $N$, $P$'s child with $N$'s left child, $N$'s left child with its right child, and right child with $P$'s right child. Under this description, one can see that two paths $((xx)x)x$ $x(x(xx))$ around the pentagon are not equal in $G$. (For comparison, the free strictly unital $A_4$-monoidal on $x$ has exactly one morphism between two trees whenever there exists such sequence of elementary moves one the other.) $G$ an groupoid which does arise from an $A_4$-monoidal one, equivalently, not from a strictly monoidal one. EDIT: As Harry notes, this not quite to your as not that it's impossible a different choice of the associators to $A_4$. It seems unlikely, but also that would be to make an elementary argument against Here is a proposed easier example, coming from C=\mathrm{Cat}$ instead of $\mathrm{Gpd}$, again with the canonical model structure. We let $C$ be the category generated by a object. So as to from $C$ a map I$ contracting a leaf which freely generates $C$ from $G$ under and naturality with respect to the elementary moves. I'll refer tensor of $p$'s and as a projection. Call canonical given by elementary moves, claim is $\beta$ associators an $A_3$-structure on $C$, that $C$'s underlying $\mathrm{Ho}(\mathrm{Cat})$-enriched category from no monoidal category. no $A_4$-monoidal structure, note that $\beta_{x,x
 sequences of elementary moveS, where an elEmentAry MovE oUt of A binAry tree $T$ is giveN By taKing a node $N$ which is the leFt chiLd OF its PArEnt $P$ aNd replaCInG $p$'S riGhT cHilD wITh $n$, $P$'s leFt cHild witH $N$'s left chiLd, $N$'S lEft child with ITs Right child, And $n$'s right child WitH $P$'s rigHt ChiLD. UndeR thIs desCriptiON, one caN see that tHe TWo diffERent patHS FrOm $((xx)X)x$ to $x(x(xx))$ around the PEnTAgon are not equaL in $G$. (FoR cOMpARIsoN, thE free stricTlY unitAL $A_4$-monoiDAl GROUpoID on $x$ has exactlY one morphisM BetWeen twO bInaRY trees WheneVeR TheRe exists any Such Sequence oF elemeNTary movES transfOrming One IntO the OThEr.) 
thuS $G$ GIveS An $a_3$-moNOidAl groupoId WhIch doEs noT ARISe frOm aN $A_4$-moNoidaL one, equivalenTly, Not fROm a StricTly moNoidAl One. 
EDiT: As HaRry noTeS, this is not quite A couNterexampLe tO yOur QuEstioN, As it's nOt cLeaR that it'S impossIBle To MAKE a Different choice of tHe ASSoCiators tHat wouLD eXtENd to $A_4$. It sEeMs uNlikELY, but aLso tHAt It would bE complICaTeD to make An ElemenTaRy aRguMent aGAinsT it. 
HerE is a propOsed eASier example, comINg from $\mathcal c=\MaTHRm{cAt}$ inSteAd of $\mathrm{GPd}$, agAIn wiTh thE CaNonICal moDel stRuCTuRE. We let $C$ be the free $A_3$-moNoIdal caTegorY generated by a Co-pointed oBJECt. So as coMparED tO $g$ from above, $C$ conTains A map $p:x\to I$ cONtractinG a leaF which frEely generATEs $C$ from $G$ UndEr $\oTimEs$ aND NaTurality with rESPect To The elemEntAry moveS. I'lL reFer To aNy Tensor proDuct of $p$'s AnD iDeNtItiEs as a PRojectioN. CAll ThE caNonicAL assocIatorS, givEn By ELemEntary mOVeS, $\ALpha$; i cLaIm thEre Is No altErnaTIve Choice $\bEta$ of assoCiaTOrs fOr An $a_3$-structUre on $C$, so that $C$'S uNderlying $\mAtHrm{ho}(\mathRM{cat})$-enricHed category arises from no MOnoidal CatEgory. 
to shOw that $C$ adMitS no $A_4$-moNoiDAl struCture, nOte thAt $\BetA_{X,X
 sequences of elementary m oves, wher e anele men ta ry m oveout of a binar y  tre e $T$ is given by taki ng ano d e $N $  w hichis thel ef t chi ld  o f i ts pa rent$P$  and re placing $P $'s  r ight child w i th  $N$, $P$' s l eft child wi th$N$'sle ftc hild,  $N $'s l eft ch i ld wit h its rig ht child, and $N$ ' s  r ight  child with $P$'s ri g ht child. Unde r this  d e sc r i pti on,  one can s ee  that the two di f f e ren t  paths from $ ((xx)x)x$ t o  $x (x(xx) )$  ar o und th e pen ta g onare not equ al i n $G$. (F or com p arison, the fre e stri ctl y u nita l  $ A_ 4$- mo n oid a lgro u poi d on $x$  h as  exac tlyo n e morp his m be tween  two binary t ree s wh e nev er th ere e xist sany s uch se quenc eof elementary m oves  transfor min gone  i nto t h e othe r.)  
T hus $G$  givesa n $ A_ 3 $ - mo noidal groupoid wh ic h do es not a rise f r om  a n  $A_4$-m on oid al o n e , equ ival e nt ly, notfrom a st ri ctly mo no idal o ne . EDI T: As Harr y note s, thisis no t  quite a count e rexample to y o ur q ue s tion , a s it's notclea r  tha t it ' simp o ssibl e toma k ea  different choice o fthe as socia tors that wou ld extendt o $A_4$. I t se e ms unlikely, butalsothat it wo u ld be co mplic ated tomake an e l e mentaryarg ume ntaga i n st  it. 
Here is a  pro po sed eas ier  exampl e,com ing  fr om  $\mathca l C=\mat hr m{ Ca t} $ i nstea d  of $\ma th rm{ Gp d}$ , aga i n with  thecano ni ca l  mo del str u ct u r e. W ele t $C $ b ethe f ree$ A_3 $-monoi dal categ ory gene ra te d by aco-pointed ob je ct. So asco mpa red to $ G$ fromabove, $C$ contains a m a p $p:x\ toI$ co ntra cting a l eaf  which  fr e ely ge nerate s $C$  f rom $ G$ un d e r$\o ti mes$ and n a t ura litywi th r espectto the elementarym ove s. I'll refer  to  any t en sor pr o duc to f $ p $ 's and identiti es as a pr oj e ct ion. Callt heca nonical  associ ators ,  givenby elemen tary move s,  $\a l p ha$ ; I claimthere is  no alter n ative ch oice$\b eta$ o fass ociat ors fo r  an  $A_3 $-stru ct ure on  $C$,  s o that $ C$'s underlying $\mathr m{Ho}( \math rm{ Cat})$-en ric h edcategoryaris es from no  mo noi dal c ate g ory.
Tos ho w t h at $C $ ad m its no $A _ 4$ -mo n o id al structur e , not e tha t $ \ beta_{ x,x
 sequences_of elementary_moves, where an elementary_move out_of_a binary_tree_$T$ is given_by taking a_node $N$ which is_the left child_of_its parent $P$ and replacing $P$'s right child with $N$, $P$'s left child with_$N$'s_left child,_$N$'s_left_child with its right child,_and $N$'s right child with_$P$'s right_child. Under this description, one can see that_the_two different paths_from $((xx)x)x$ to $x(x(xx))$ around the pentagon are not_equal in $G$. (For comparison, the_free strictly unital_$A_4$-monoidal_groupoid_on $x$ has exactly_one morphism between two binary trees_whenever there exists any such sequence_of elementary moves transforming one into the_other.) 
Thus $G$ gives an $A_3$-monoidal_groupoid which does not arise_from an_$A_4$-monoidal one, equivalently, not from_a strictly monoidal_one. 
EDIT:_As Harry notes,_this is not quite a counterexample_to your question,_as it's not clear that it's_impossible_to make a_different_choice_of the_associators that would_extend_to $A_4$._It_seems unlikely, but also that it_would_be complicated to make an elementary argument_against it. 
Here is_a_proposed easier example, coming_from $\mathcal C=\mathrm{Cat}$ instead of_$\mathrm{Gpd}$, again with the canonical model_structure. We_let $C$_be the free $A_3$-monoidal category generated by a co-pointed object. So_as compared to $G$ from above,_$C$ contains a map_$p:x\to I$_contracting_a leaf which_freely_generates $C$_from $G$ under $\otimes$ and naturality with_respect to_the elementary moves. I'll refer to_any tensor product of_$p$'s_and identities as a projection. Call_the canonical associators, given by elementary_moves, $\alpha$; I claim there_is_no_alternative choice $\beta$ of associators_for an $A_3$-structure on $C$, so_that $C$'s underlying_$\mathrm{Ho}(\mathrm{Cat})$-enriched category arises from no monoidal category._
To_show that $C$ admits no $A_4$-monoidal_structure,_note that $\beta_{x,x