Old Work Box
For cutouts in plywood, as opposed to frame mounted receptacles.
Old work box in 60 seconds:
Note on dimensions - the outlet mounting screws form the outer edge. These screws are protrusions, or inside the box, but their outer dimension defines the height of the box. That appears to be 3-5/8". It is safe to do 3-3/4" as the 4 front tabs are larger than this. The locking tabs on the back are even larger.
It seems safe that 3.75 is a safe height of cutout, and 2-3/8" is a safe width (widths all appear to be 2.25").
- 3-7/8-Inch Length by 2-3/8 
- 4-1/8 x 2-1/4 
- 2-3/4 in x 2-1/4 in 
- 2.75 x 2.25 x 4.3 
- 3.65 x 2.3 x 4.3 
- 2.25 x 4.38 x 2.75 
Allowed in New Construction
Just a heads up, if the wall is fire rated the box must be secured to the stud regardless of the box listing. This is a building code issue not a electrical code issue.
I have no idea if the opening posters wall is fire rated.