453,021,789,719,303,692,337 face center positions can be achieved. If like-colored face centers are not distinguished, the best lower bound I can find using this method is three hypermoves. If stomach cubies are considered, I think both bounds increase by one, since only deep moves can touch them. It seems strange that this method relies only on the face center solution. Similar arguments about edges are not as good, because so many edge positions are achievable using shallow hypermoves.
Corners are practically irrelevant, since they can be fixed using only shallow hypermoves.