Imagine an $n \times n \times n$ Rubik's cube, where we can transition the state of the cube using Singmaster moves under either the face- or quarter-turn metrics. We call a face of this cube "solved" if all of the symbols on the face are of the same color (there are six total colors).

How many total cube states are there when $k = {0, 1, 2, 3, 4}$ of the cube faces are solved?

Note: This is (hopefully) a simplification of an earlier question asking for the probability that a greedy algorithm solves a Rubik's cube, where once a face is solved, the algorithm cannot backtrack and perturb the face (though rotations of the face are allowed).