Random-graph conjecture on square-free spanning-tree counts

Let $\EuScript{G}(n,p)$ be a random simple undirected graph with $n$ vertices and edge-probability $0\leq p\leq 1$. For $c>1$, assume $c\log(n)/n<p(n)<1-o(\log(n)/n)$. Square-free complexity conjecture. As $n\to\infty$, the probability that $\kappa(\EuScript{G}(n,p))$ is square-free is $(1-o(1))6/\pi^{2}$. Since a connected undirected graph with square-free $\kappa(G)$ has cyclic (or trivial) critical group, this conjecture is intended to explain the preceding cyclicity prediction using the asymptotic density of square-free integers. Its status is not resolved in the source.