Wanless's conjecture on transversals in latin hypercubes

A latin hypercube of order $n$ and dimension $d$ is a $d$-dimensional array filled with $n$ symbols so that every line contains all symbols exactly once. A transversal is a diagonal containing all $n$ symbols. Wanless's conjecture. Every latin hypercube of odd order or odd dimension has a transversal.

The conjecture is trivial for order $2$ and easy to prove for order $3$. For order $4$, all latin hypercubes have transversals except the hypercube described as $\mathcal{Q}_4^d$ when $d$ is even; all latin hypercubes of order $5$ have transversals. The conjecture is also stated to have been extended from multidimensional permutations to all polystochastic matrices.