Virtual specialness conjecture for periodic generalized Bestvina–Brady groups

Let $L$ be a flag complex, let $M$ be a covering of $L$, let $S\subseteq\mathbb Z$ satisfy $S+n=S$ for some $n>0$, and let $G_L^M(S)\rtimes n\mathbb Z$ be the associated CAT(0) cubical group. Write $\pi(M,L)$ for the group associated to the covering $M\to L$. Virtual specialness conjecture. If $\pi(M,L)$ is finite and $S+n=S$ for some $n>0$, then the CAT(0) cubical group $G_L^M(S)\rtimes n\mathbb Z$ is virtually special. The source explains that this group acts properly and cocompactly on the relevant CAT(0) cube complex, and proposes virtual specialness in this periodic finite-covering case; no resolution is supplied.