Enhanced Cohen–Lenstra distribution conjecture

Let $\mathbf{n}=(n_1,\ldots,n_k)$ be a tuple of distinct positive integers, and let $S$ be a finite collection of primes not containing $p$. For a geometric family $\mathcal{F}_g$ of abelian varieties over $\mathbb{F}_p$, suppose that the Cohen–Lenstra distribution $\mathrm{CL}_S$ accurately models the distribution of $A(\mathbb{F}_p)_S$ for randomly sampled $A\in\mathcal{F}_g$ as $g$ grows.

Enhanced Cohen–Lenstra conjecture. The $\mathbf{n}$-Cohen–Lenstra distribution $\mathrm{CL}_{\mathbf{n},S}$ accurately models the joint distribution of

$$A(\mathbb{F}_{p^{n_1}})_S,\ldots,A(\mathbb{F}_{p^{n_k}})_S$$

for randomly sampled $A\in\mathcal{F}_g$ as $g$ grows.

This conjecture extends the Cohen–Lenstra heuristic from the group of rational points over $\mathbb{F}_p$ to joint distributions over several finite extensions. The source gives no evidence of a resolution.