Transversal conjecture for Latin hypercubes of odd order or dimension

Let $M(d,n)$ denote the set of Latin hypercubes of dimension $d$ and order $n$. A transversal is a set of $n$ entries containing each value in every coordinate position and each symbol exactly once. Transversal conjecture. If $n$ is odd or $d$ is odd and $H\in M(d,n)$, then $H$ has transversals. The conjecture contrasts with the theorem that no transversals exist in $\mathbb{Z}_n^d$ when both $n$ and $d$ are even; its status is not resolved in the supplied text.