Lemon-Limify a String Write a program or function that takes in a nonempty single-line string. You may assume it only contains printable ASCII excluding space.

Print or return an ASCII art lozenge shape similar to a lemon or lime made from the prefixes of the string.

Suppose the input string is n letters long. Then, such a shape consists of 2n&nbsp;&minus;&nbsp;1 columns of ASCII art stitched together, each consisting of 2n&nbsp;&minus;&nbsp;1 lines. Counting from 1, the k-th column is f(k)&nbsp;=&nbsp;min(k,&nbsp;2n&nbsp;&minus;&nbsp;k) characters wide, and contains f(k) copies of the first f(k) characters of input, centered vertically, with single blank lines separating the copies.

For example, if the input is Lemon, the output should be:

          Lemon
      Lemo     Lemo
   Lem    Lemon    Lem
 Le   Lemo     Lemo   Le
L  Lem    Lemon    Lem  L
 Le   Lemo     Lemo   Le
   Lem    Lemon    Lem
      Lemo     Lemo
          Lemon


If the input is lime the output should be:

      lime
   lim    lim
 li   lime   li
l  lim    lim  l
 li   lime   li
   lim    lim
      lime


And the same pattern is followed for other inputs:

a

a

Be

 Be
B  B
 Be

/\

 /\
/  /
 /\

cat

   cat
 ca   ca
c  cat  c
 ca   ca
   cat

|||

   |||
 ||   ||
|  |||  |
 ||   ||
   |||

.__.

      .__.
   .__    .__
 ._   .__.   ._
.  .__    .__  . 
 ._   .__.   ._
   .__    .__
      .__.

$tring

               $tring
          $trin      $trin
      $tri     $tring     $tri
   $tr    $trin      $trin    $tr
 $t   $tri     $tring     $tri   $t
$  $tr    $trin      $trin    $tr  $
 $t   $tri     $tring     $tri   $t
   $tr    $trin      $trin    $tr
      $tri     $tring     $tri
          $trin      $trin
               $tring


Lines in the output may have trailing spaces and there may be one optional trailing newline.

The shortest code in bytes wins.
Matlab, 140 136 128 124 bytes
Basically first starts with the middle section, and then prepends/appends the shortened/modified versions step by step.
a=input('');v=ones(nnz(a)*2-1,1)*a;v(2:2:end,:)=0;b=v;for k=a;v=v(2:end,1:end-1);v(end+1,:)=0;b=[v,b,v,''];end;b(~flip(b))=0

Thanks for 8 bytes @LuisMendo!
E.g. for MATLAB we get:
               MATLAB               
          MATLA      MATLA          
      MATL     MATLAB     MATL      
   MAT    MATLA      MATLA    MAT   
 MA   MATL     MATLAB     MATL   MA 
M  MAT    MATLA      MATLA    MAT  M
 MA   MATL     MATLAB     MATL   MA 
   MAT    MATLA      MATLA    MAT   
      MATL     MATLAB     MATL      
          MATLA      MATLA          
               MATLAB