Cyclicity-index conjecture for imaginary quadratic class groups

Fix $n\in\mathbb{N}$ and a partition $\lambda=(n_1,\dots,n_r)$ of $n$, with $n_1\geq\cdots\geq n_r\geq1$, and define

$$G_\lambda(p)=\bigoplus_{i=1}^r\mathbb{Z}/p^{n_i}\mathbb{Z},\qquad \operatorname{c}(\lambda)=\sum_{i=1}^r(3-2i)n_i.$$

For a finite abelian group $G$, let $\mathcal{F}(G)$ denote the number of negative fundamental discriminants whose imaginary quadratic class group is isomorphic to $G$. Cyclicity-index conjecture. The inequality $\mathcal{F}(G_\lambda(p))>0$ holds for infinitely many primes $p$ if and only if $\operatorname{c}(\lambda)\geq0$. If $\operatorname{c}(\lambda)>0$, then

$$\mathcal{F}(G_\lambda(p))\sim\frac{\mathfrak{C}}{n}\frac{p^{\operatorname{c}(\lambda)}}{\log p}$$

as $p\to\infty$. If $\operatorname{c}(\lambda)=0$, then

$$\sum_{\substack{p\leq x\p\text{ prime}}}\mathcal{F}(G_\lambda(p))\sim\frac{\mathfrak{C}}{n}\frac{x}{(\log x)^2}.$$

If $\operatorname{c}(\lambda)<0$, then $p\gg_\lambda1$ implies $\mathcal{F}(G_\lambda(p))=0$. This gives a precise predicted criterion for which abelian $p$-group shapes occur for infinitely many primes, but the source does not specify a resolution.