Alspach et al.'s wreath-product conjecture for hamiltonian decomposable digraphs

Let $G$ and $H$ be hamiltonian decomposable directed graphs. Their wreath product $G \wr H$ is the digraph on vertex set $V(G)\times V(H)$ with an arc from $(g_1,h_1)$ to $(g_2,h_2)$ if and only if there is an arc from $g_1$ to $g_2$ in $G$, or there is an arc from $h_1$ to $h_2$ in $H$ and $g_1=g_2$. A directed graph is hamiltonian decomposable if it admits a decomposition into directed hamiltonian cycles.

Alspach et al.'s wreath-product conjecture. If $G$ and $H$ are hamiltonian decomposable directed graphs, then $G\wr H$ is also hamiltonian decomposable.

This is a long-standing conjecture for directed graphs, first appearing in Alspach et al. (1987). The paper affirms most of its previously open cases, but the supplied text does not establish that every case has been resolved.