Virtual torsion-freeness conjecture for generalized Bestvina–Brady groups

Let $L$ be a flag complex, let $M$ be a covering of $L$, let $S\subseteq\mathbb Z$ be periodic, and let $G_L^M(S)$ denote the associated generalized Bestvina–Brady group. Write $\pi(M,L)$ for the group associated to the covering $M\to L$. Virtual torsion-freeness conjecture. If $S$ is periodic and $\pi(M,L)$ is virtually torsion-free, then $G_L^M(S)$ is virtually torsion-free. This conjecture asserts that the corresponding necessary condition is also sufficient; the source notes possible topological obstructions but gives no counterexamples and identifies the universal-cover case over a flag triangulation of $\mathbb R P^2$ as an important test case.