A Megaminx is a dodecahedral twisty puzzle similar to a Rubik's cube. Each of the twelve faces has a different color; the puzzle is cut into 50 pieces by slices parallel to the faces, which may be turned to align to different faces, just like a Rubik's cube. Each face contains 11 pieces: one center, five edges, and five corners. There are 12 colors; is there a sequence of moves such that no face in the resulting configuration contains repeated colors?
If we consider the question of just coloring each of the faces of the pieces (i.e. peeling off the stickers and putting them back on), it is not hard to show that the answer is yes. However, there are two forms of restriction that make this question more complicated. The first is that only certain combinations of colors can appear on the same piece: e.g. colors corresponding to opposite centers can never be on the same piece, and no two pieces can have the same color scheme. Is there a coloring that satisfies the desired property that is a rearrangement of the pieces of a Megaminx?
The second complication is that, due to parity constraints, certain positions cannot be reached by valid moves, even if all the pieces are properly colored. For example, it is impossible to rotate only a single corner piece by $120^\circ$ (more precisely, only even permutations of corners and edges are possible). Is there such a coloring that can be reached by a sequence of valid moves?
I realize that this could be answered by essentially brute-force running cases through a Megaminx solver, but that approach seems not very mathematically interesting.