Cohen–Lenstra conjecture for class-group $p$-parts of number fields

Let $K$ be a number field of degree $n$, let $D_K$ be its discriminant, and let $Cl_{K,p}$ be the $p$-part of its class group. Let $S$ be the set of finite abelian $p$-groups, and for $A\re S$ define

$$S_X^{7}(A)=\frac{\left|\left\{K\mid 0<7 D_K<X,\ Cl_{K,p}\cong A\right\}\right|}{\sum_{K,0<7 D_K<X}1}.$$

For $s\in\mathbb{Z}_{\geq 1}\cup\{\infty\}$, set

$$\eta_s(p)=\prod_{i=1}^s(1-1/p^i),$$

and define the CohenLenstra probability measure

$$\mu_u(A)=\frac{\eta_\infty(p)}{|\operatorname{Aut}A|\,|A|^u}.$$

CohenLenstra conjecture. For every $A\in S$,

$$\mu_0(A)=\lim_{X\longrightarrow\infty}S_X^-(A),$$ $$\mu_1(A)=\lim_{X\longrightarrow\infty}S_X^+(A).$$

The conjecture predicts limiting distributions for the $p$-parts of class groups, with separate measures for negative and positive discriminants. The source gives no resolution status for this general formulation; it notes that the original work treated quadratic fields with $p\neq 2$ separately by signature.