Frequency-measure conjecture for p-class tower groups

Let $G$ be a finitely generated pro-$p$ group. Let $\operatorname{Freq}(G)$ denote its limiting frequency among imaginary quadratic fields, and let $\operatorname{Meas}(G)$ denote its Cohen–Lenstra-type measure in the category of Schur groups. Frequency-measure conjecture. For every pro-$p$ group $G$,

$$\operatorname{Freq}(G)=\operatorname{Meas}(G).$$

In particular, $\operatorname{Freq}(G)=0$ if $G$ is not Schur; for finite Schur groups, the frequency is given by the formula stated elsewhere in the paper. This is the paper’s main conjectural identification of arithmetic frequencies with group-theoretic measures.