Nonexistence of a quadrisection of the six-dimensional torus by subcube partitions

Let $T^6=(\mathbb{R}/4\mathbb{Z})^6=[0,4]^6/\sim$, and partition its $4^6$ subcubes into four classes. Nonexistence conjecture. No partition of the subcubes of $[0,4]^6/\sim$ gives a quadrisection of $T^6$. The paper explains that the symmetric construction used in odd dimensions fails because $4$ and $6$ are not relatively prime. It suggests that a less symmetric partition might exist but reports that trial and error indicates this is unlikely; no proof of nonexistence is given.