Harrington et al.'s divisibility conjecture for modulo-dependent chip-collecting games

Let $m$ and $n$ be positive integers with $m\mid n$, and let $a<b<\min\{m,n\}$. Consider the random walk on $\mathbb{Z}_m\times\mathbb{Z}_n$ whose moves are $(+a,+b)$ and $(+b,+a)$, with winning positions determined by the game. Harrington et al.'s divisibility conjecture. If all winning positions are of the form $(0,y)$, then

$$m\mid(b^2-a^2).$$

This conjecture concerns when the winning positions in the extended modulo-dependent game have a prescribed form. The supplied source does not state whether it has been resolved.