Exterior Stanley–Reisner cohomology conjecture for oriented pro-p RAAGs

Let $\Gamma=(\mathcal{V},\mathcal{E})$ be a special digraph, let $G$ be the oriented pro-$p$ right-angled Artin group associated to $\Gamma$, and let $q$ be a $p$-power with $q\neq2$ when $p=2$. Let $\mathbf{\Lambda}_\bullet(\Gamma^\ast)$ denote the exterior Stanley–Reisner $\mathbb{F}_p$-algebra associated to $\Gamma$. Exterior Stanley–Reisner cohomology conjecture. One has

$$\mathbf{H}^\bullet(G)\simeq\mathbf{\Lambda}_\bullet(\Gamma^\ast).$$

This is the oriented version of the conjecture cited from QSV. Determining which digraphs have this cohomological property remains an open problem.