Solvability of uncountable edgeless cubes

Determine whether every convergent scramble of an edgeless infinitary Rubik’s cube whose layer set has uncountable cardinality is solvable in principle.

Background

The paper proves that every convergent scramble of the countable edgeless cube is solvable in at most $\omega^2$ moves, while the corresponding uncountable case is not addressed. The authors identify the uncountable edgeless cube as a principal unresolved extension of their countable algorithm.