The sharp random-graph threshold for relative hyperbolicity and thickness

Let $\mathcal{G}_{n,p}$ denote the Erdős–Rényi random graph, let $\Gamma\sim\mathcal{G}_{n,p}$, and let $W_{\Gamma}$ be the right-angled Coxeter group with presentation graph $\Gamma$. A group is thick of order $k$ if it has thickness order $k$, and $a.a.s.$ means asymptotically almost surely. The sharp-threshold conjecture. The following hold: (i) if $p=p(n)=o\left(\frac{1}{\sqrt{n}}\right)$, then $a.a.s.$ $W_{\Gamma}$ is relatively hyperbolic; (ii) if $p=p(n)=\Omega\left(\frac{1}{\sqrt{n}}\right)$, then $a.a.s.$ $W_{\Gamma}$ is thick of order $O(1)$ and has polynomial divergence. This would sharpen the paper's threshold theorem. A key unresolved issue is whether, below the threshold, a random graph can have finite thickness order tending to infinity with $n$; the source provides no resolution status.