This seems like a deceptively simple problem. Quite tempted to purchase a set to try my hand at it and fail miserably. I am finding kits available for $40-50.

I am also wondering if there was a clue the author did not mention or ignored. Will a final solution have a checkerboard on all sides? If so, that seems a potentially significant simplification to the problem.

Is it possible to distort the y pentamino shape in a way that prevents those 3 swappy configurations?

Interestingly, first 3 solutions have a pack of 5 identically oriented pieces on one side of the cube, but not the other two (the last solution also has a pack of 5 on the side, but with one piece flipped).