I attempted that at one point, and perhaps there's a clever way to get that solution to work, but I wasn't able to figure it out for my use case.
Part of the problem is related to the way I'm doing rotations on the 26 three.js objects that make up the cube. It's MUCH simpler if the cubes that make up the puzzle have a rotation of (0,0,0) at the start of every move (simpler for me to understand at least).
You can see the work in progress here www.solvethecube.ca
It's just a little vanity project, but it makes heavy use of gsap.