The connectivity conjecture for the crossing graph of random right-angled Coxeter groups

Let $\Gamma$ be a random graph in $\mathcal{G}(n,p)$, and let $\square(\Gamma)$ denote the graph whose vertices are induced Morse $4$-cycles of $\Gamma$, with adjacency determined by the crossing relation.

Connectivity conjecture. For every $\epsilon>0$, if

$${{\overline A}^{+}({(})}n)>(1+\epsilon)\sqrt{\frac{\log n}{n}},$$

then, asymptotically almost surely, $\square(\Gamma)$ is connected.

This conjecture identifies the proposed sharp threshold for connectedness of $\square(\Gamma)$, complementing the result that below the corresponding lower threshold $\square(\Gamma)$ has an isolated vertex. The source does not provide a resolution of the conjecture.