showed that any tiling that forms a counterexample to the conjecture can be assumed to have an even more special form: the cubes have side length a power of two and integer vertex coordinates, and the tiling is periodic with period twice the side length of the cubes in each coordinate direction. Based on this geometric simplification, he also simplified Hajós' group-theoretic formulation, showing that it is sufficient to consider abelian groups that are the direct sums of cyclic groups of order four, with each .
