Arc-disjoint hamiltonian paths in Cartesian products of directed cycles

Let $C_1,C_2,\ldots,C_r$ be directed cycles of length at least $2$, and let $r\geq 2$. Their Cartesian product is the digraph

$$C_1\mathbin{\Box}C_2\mathbin{\Box}\cdots\mathbin{\Box}C_r.$$

Arc-disjoint hamiltonian paths conjecture. The Cartesian product $C_1\mathbin{\Box}C_2\mathbin{\Box}\cdots\mathbin{\Box}C_r$ has two arc-disjoint hamiltonian paths.

The claim is proved for $r=2$ and for $r\geq 4$ in the paper, leaving the case of Cartesian products of three directed cycles as the only open case. Several broad families of three-cycle products are also known to satisfy the claim.