In the paper [1] they do discuss generalizing to larger boards. Puzzle's like the 2D slide puzzle or 3D Rubick's cube are modeled by different permutation groups. So, their techniques will work for the Rubick's cube as well.

They likely wrote the paper the way they did because (1) Rubick's cube math is much more convoluted to explain, just because it is much larger, (2) Just working out the matrix representation for large groups is a pain. I know because I worked on this sort of representation theory in my PhD, and in fact, I thought of a very similar idea back then but never got around to writing a paper. Partially because nice implementations of representation theory algorithms do (or at least did) not exist.

[1] https://arxiv.org/pdf/2410.22287