Clancy–Leake–Payne conjecture on cyclic critical groups of random graphs

Let $G$ be a graph on $n$ vertices, and let $K(G)$ denote its critical group. Consider the proportion of connected graphs with $n$ vertices whose critical group is cyclic.

Clancy–Leake–Payne conjecture. As $n$ tends to infinity,

$$\lim_{n \to \infty} \frac{\#\{\text{Connected graphs } G \text{ with } |V(G)| = n \text{ and } K(G) \text{ cyclic}\}}{2^{\binom{n}{2}}} = \zeta(3)^{-1}\zeta(5)^{-1}\zeta(7)^{-1}\zeta(9)^{-1}\zeta(11)^{-1}\cdots \approx 0.7935212.$$

This conjecture gives the limiting probability that the critical group of a uniformly random graph is cyclic, a question motivated by the fact that the critical group of a graph is typically large. The source presents it as a conjecture and gives no resolution evidence.