I could see getting a psoas stretch on the opposite side from the one usually considered targeted by pigeon... that is, traditional pigeon hip-rotator stretch on the bent-leg side, psoas stretch on the extended leg side.

If my speculation is the case, the variation lying on the back probably wouldn't give that same benefit, because it removes the hip extension; both sides are in flexion to different degrees, and at different rotations.
But you could look for other shapes that give hip extension.