Sawin–Wood conjecture for class groups of Gamma-extensions containing roots of unity

Let $\Gamma$ be a finite group and let $p$ be a prime with $p\nmid |\Gamma|$. Let

$$S=\mathbb{Z}_p[\Gamma]/\left(\sum_{\gamma\in\Gamma}\gamma\right).$$

Let $K_0$ be a number field with $u$ infinite places containing the $p^r$th roots of unity but not the $p^{r+1}$th roots of unity, for some $r\geq 1$. Let $\mathcal{E}=\mathcal{E}(\Gamma,K_0)$ be the set of isomorphism classes of Galois $\Gamma$-extensions $K/K_0$ together with an isomorphism $\operatorname{Gal}(K/K_0)\simeq\Gamma$, and write $\operatorname{Cl}_{K|K_0}=\operatorname{Cl}_K/\operatorname{Cl}_{K_0}$. Sawin–Wood conjecture. As $K$ varies over $\mathcal{E}$, the distribution of $S$-modules $\operatorname{Cl}_{K|K_0}[p^\infty]$ has average number of surjective morphisms to any finite $S$-module $V$ equal to

$$\frac{|(\wedge^2_{\mathbb{Z}_p}V)^\Gamma[p^r]|}{|V|^u}.$$

The claimed moments determine a unique distribution on $S$-modules, for which the paper gives an explicit formula. This is a conjectural refinement of Cohen–Lenstra–Martinet heuristics at primes dividing the order of roots of unity in the base field.