Malle's moment predictions for the 2-class groups of cubic fields

Let $K$) run through all isomorphism classes of cubic fields ordered by discriminant, and let $\operatorname{Cl}_2(K)$ denote the $2$-class group of $K$. A cubic field is totally real if it has three real embeddings and complex if it has one real embedding and one pair of complex-conjugate embeddings.

Malle's conjecture. The average size of $\operatorname{Cl}_2(K)$ over totally real cubic fields $K$ is $5/4$, and its second moment is $15/8$; over complex cubic fields $K$, the average size is $3/2$, and its second moment is $3$.

These predictions arise from Cohen–Lenstra and Cohen–Martinet class-group heuristics adjusted for the presence of second roots of unity in the base field. The claims about the average sizes were proven by Bhargava; the second-moment assertions are the focus of the paper and, in the supplied context, are not stated as resolved.