Density conjecture for primes in the rational 2-Selmer group of $S_n$-fields

Let $p$ be a prime and let $n=2m$. Write $\operatorname{Sel}_2^{\mathbb{Q}}(F)$ for the rational part of the 2-Selmer group of an $S_n$-number field $F$, and let $c(n,p)$ denote the local mass defined by the weighted count of degree-$n$ $\mathbb{Q}_p$-algebras. The rational 2-Selmer prime conjecture. As $F$ varies over $S_n$-number fields of degree $2m$ with signature $(r_1,r_2)$ ordered by absolute discriminant,

$$\operatorname{Prob}(F\mid p\in\operatorname{Sel}_2^{\mathbb{Q}}(F))=\frac{c(m,p)}{c(2m,p)}\cdot\frac{1}{p^m}.$$

This is a local-to-global prediction for the rational 2-Selmer group and underlies the paper's predictions for rationally imprimitive types; it is not known in general.