Hamiltonian-path distance congruence conjecture for Cartesian products of directed cycles

Let $X$ be the Cartesian product of $k$ directed cycles of lengths $m_1,m_2,\ldots,m_k$, where $k\geq 3$ and each $m_i\geq 2$. For vertices $u$ and $v$ of $X$, let $d_X(u,v)$ denote the length of a shortest directed path from $u$ to $v$. Hamiltonian-path distance congruence conjecture. There is a Hamiltonian path from $u$ to $v$ if and only if

$$d_X(u,v)\equiv -1\pmod{\gcd(m_1,m_2,\ldots,m_k)}.$$

This generalizes the coprime-length case, in which the congruence condition is automatic, and remains open according to the supplied source.