Successive-maxima conjecture for non-genus class numbers of abelian fields

Let $G$ be an abelian group and let ${\mathcal K}_G^s$ be the family of abelian fields with Galois group $G$ and signature $s$. For a conductor $f$, let ${\mathcal F}_{G,f}^s$ consist of fields in this family with conductor $f$, and let ${\mathcal F}'{}_{G,f}^s$ consist of fields with conductor dividing $f$. For $K$ in the family, write $D_K$ for its discriminant, $g_{K/\mathbb Q}$ for its genus number, and $\operatorname{Cl}_K$ for its class group. For $0<\varepsilon<1$, define

$$C_{\varepsilon}({\mathcal F}_{G,f}^s)=\left[\prod_{K\in{\mathcal F}_{G,f}^s}\frac{|\operatorname{Cl}_K|}{g_{K/\mathbb Q}(\sqrt{|D_K|})^{\varepsilon}}\right]^{1/|{\mathcal F}_{G,f}^s|},$$

with analogous definitions of $C'_{\varepsilon}({\mathcal F}'{}_{G,f}^s)$ and $C_{\varepsilon}(K)$. Let $r(G)$ be the minimal number of generators of $G$. If $|G|=\prod_{i=1}^t p_i^{n_i}$, put $R(G)=\sum_{i=1}^t n_i$.

Successive-maxima conjecture. (i) As $f$ increases from its minimum, the successive maxima of $C_{\varepsilon}({\mathcal F}_{G,f}^s)$, $C'_{\varepsilon}({\mathcal F}'{}_{G,f}^s)$, and $C_{\varepsilon}(K)$ occur, with probability $1$, for conductors satisfying

$$r(G)\leq\omega(f)\leq R(G).$$

This property is independent of $\varepsilon$. (ii) For every $0<\varepsilon<1$, each of these sequences of successive maxima is infinite.

The probability-one qualification means that the assertion may fail on a set of conductors of density zero. The statement is motivated by computations of class numbers after removing the genus contribution; the behavior of the number of prime divisors in the indicated interval remains unclear, and the possible even-order wild-ramification exception is explicitly neglected in the source.