3 gang box, code specifications
There was a change to the 2020 box fill calculation but you only have 4 grounds so your totals will be the same.
14 gauge is a 2cu inch multiplier.
So you have 9 current carrying conductors
4 grounds= 1
Each switch is a 2x so 6 (a total of 12 cu in)
16 x2 = 32 cu inch if there are clamps add 2 = 34cu in
so you should be fine You have a 10 cu inch safety.
Added with the addition of the feed wire:
With the addition of the feed wire 14-2 this adds 4.5 to the total for 38.5 cu inch
See comments for the complete calculations.
The box fill has changed and this change for 2020 increased the calculated fill by +.5 cu inch.