  Given a graph $\Gamma$, its auxiliary \emph{square-graph} $\square(\Gamma)$
is the graph whose vertices are the non-edges of $\Gamma$ and whose edges are
the pairs of non-edges which induce a square (i.e., a $4$-cycle) in $\Gamma$.
We determine the threshold edge-probability $p=p_c(n)$ at which the Erd{\H
o}s--R\'enyi random graph $\Gamma=\Gamma_{n,p}$ begins to asymptotically almost
surely have a square-graph with a connected component whose squares together
cover all the vertices of $\Gamma_{n,p}$. We show
$p_c(n)=\sqrt{\sqrt{6}-2}/\sqrt{n}$, a polylogarithmic improvement on earlier
bounds on $p_c(n)$ due to Hagen and the authors. As a corollary, we determine
the threshold $p=p_c(n)$ at which the random right-angled Coxeter group
$W_{\Gamma_{n,p}}$ asymptotically almost surely becomes strongly algebraically
thick of order $1$ and has quadratic divergence.
