Nonabelian Cohen–Lenstra distribution conjecture for pro-prime-to-roots-of-unity class groups

Let $Q={\mathbb Q}$ or $\mathbb F_q(t)$, and let $\mu_Q$ be the group of roots of unity contained in $Q$. Let $\Gamma$ be the finite group and $\Gamma_\infty$ the cyclic subgroup governing the extensions under consideration. As $K$ varies over $(\Gamma,\Gamma_\infty)$-extensions of $Q$, consider the pro-prime-to-$|\mu_Q|$ completion of $G_{\mathcal O}^{\#}(K)$. For an admissible $\Gamma$-group $H$ with $\gcd(|H|,|\mu_Q|)=1$ and $\operatorname{char}(Q)\nmid |H|$, let its $H$-moment be the expected number of $\Gamma$-equivariant surjections onto $H$. Nonabelian Cohen–Lenstra conjecture. The distribution of the pro-prime-to-$|\mu_Q|$ completion of $G_{\mathcal O}^{\#}(K)$ is predicted by the probability measure $\mu_{\Gamma,\Gamma_\infty}$ defined by the random group model. Moreover, the $H$-moment of the distribution of $G_{\mathcal O}^{\#}(K)$ is

$$\frac{1}{[H^{\Gamma_\infty}:H^{\Gamma}]}.$$

This is the main conjectural extension of the Cohen–Lenstra heuristics to nonabelian admissible $\Gamma$-groups. The source gives no evidence that the distribution statement or the stated moments are resolved in the generality claimed.