Delaunay–Jouhet moment conjecture for class groups of quadratic fields

Let $p$ be a prime with $p\geqslant 3$. For a finite abelian $p$-group of type $\lambda$, let $C_{\lambda,\mu}(p)$ denote the number of subgroups of type $\mu$. For a quadratic field $K_d$ of fundamental discriminant $d$, write $\operatorname{C\ell}(K_d)[p^j]$ for the subgroup annihilated by $p^j$.

Delaunay–Jouhet moment conjecture. For every positive integer $\ell$ and partition $\lambda=1^{m_1}2^{m_2}\cdots \ell^{m_\ell}$, the average over fundamental negative discriminants satisfies

$$\operatorname{Avg}_d\,\prod_{j=1}^{\ell}|\operatorname{C\ell}(K_d)[p^j]|^{m_j}=\sum_{\mu\subseteq\lambda}C_{\lambda,\mu}(p),$$

while the average over fundamental positive discriminants satisfies

$$\operatorname{Avg}_d\,\prod_{j=1}^{\ell}|\operatorname{C\ell}(K_d)[p^j]|^{m_j}=\sum_{\mu\subseteq\lambda}C_{\lambda,\mu}(p)p^{-|\mu|}.$$

Here the sums range over integer partitions $\mu\subseteq\lambda$.

This is a moment formulation of the Cohen–Lenstra heuristics for the $p$-primary parts of class groups of quadratic fields. The source presents it as a conjecture and gives no resolution of the full statement.