Clancy–Leake–Payne conjecture on paired Sylow subgroups of random graph critical groups

Fix $0<q<1$. Let $G(n,q)$ be an Erdős–Rényi random graph, let $p$ be a prime, and let $\Gamma$ be a finite abelian $p$-group equipped with a perfect pairing $\langle\cdot,\cdot\rangle$. Let $K(G(n,q))_{(p)}$ denote the Sylow $p$-subgroup of the critical group, equipped with its associated monodromy pairing. For a paired group, write $\operatorname{Aut}(\Gamma,\langle\cdot,\cdot\rangle)$ for the automorphisms preserving the pairing.

Clancy–Leake–Payne conjecture. As $n$ tends to infinity, the probability that $K(G(n,q))_{(p)}$ with its monodromy pairing is isomorphic to $(\Gamma,\langle\cdot,\cdot\rangle)$ is

$$\frac{\prod_{i=1}^{\infty}(1-p^{1-2i})}{|\Gamma|\,|\operatorname{Aut}(\Gamma,\langle\cdot,\cdot\rangle)|}.$$

Here $\operatorname{Aut}(\Gamma,\langle\cdot,\cdot\rangle)$ is the set of automorphisms of $\Gamma$ that preserve the pairing. This is the paired analogue of the distribution theorem for Sylow subgroups of critical groups of Erdős–Rényi random graphs; the source gives no resolution evidence.