Scenario one. You have an 8" midbass driver in a box
crossed over from 70hz to 500hz.

Scenario two. You desire to replace that 8" inch driver with a
12" midbass driver but don't have the space on the front baffle of
the box because there is only 8" width available, so you compromise
a little bit and you recess the 12" midbass ~6" to 12" inside the box
and taper the wall so the energy squirts out of the 8" opening on
the front baffle.

Questions are;

How much senstivity will you lose, isn't this sorta like a reverse
horn effect sensitivity-wise?

How much more "punch" in the music do you gain, if any, vs. the 8"
driver?