Maximum-transversal conjecture for order-four latin hypercubes

Let $Q^n(\mathbb{Z}_4)$ and $Q^n(\mathbb{Z}_2^2)$ denote the $n$-dimensional latin hypercubes isotopic to the Cayley tables of the corresponding iterated groups. A transversal is a set of $4$ entries containing all distinct symbols and lying pairwise at Hamming distance $n$. Maximum-transversal conjecture. Among all latin hypercubes of order $4$, the maximum number of transversals is attained by the odd-dimensional Cayley tables of the iterated group $\mathbb{Z}_4$ and by the Cayley tables of the iterated group $\mathbb{Z}_2^2$. The paper establishes exact counts for these group-derived hypercubes and characterizes the order-four hypercubes with no transversals, but the asserted global maximality remains conjectural.