Bhargava's local specification density conjecture for $S_n$-number fields

Let $n$ be a positive integer, and let $\Sigma=(\Sigma_2,\Sigma_3,\Sigma_5,\ldots)$ be an acceptable collection of local specifications, where each $\Sigma_p$ is a set of $\mathbb{Q}_p$-algebras. For a local algebra $F_p$, write $\operatorname{Disc}(F_p)$ for its discriminant and $\operatorname{Aut}_{\mathbb{Q}_p}(F_p)$ for its automorphism group. Bhargava's conjecture. As $F$ varies over $S_n$-number fields with fixed signature $(r_1,r_2)$ ordered by absolute discriminant,

$$\operatorname{Prob}(F\mid F\otimes_{\mathbb{Q}}\mathbb{Q}_p\in\Sigma_p)=\prod_p\frac{1}{c(n,p)}\left(\sum_{F_p\in\Sigma_p}\frac{1}{\operatorname{Disc}(F_p)}\cdot\frac{1}{\#\operatorname{Aut}_{\mathbb{Q}_p}(F_p)}\right).$$

This predicts independence of local specifications with the mass prescribed by discriminants and automorphism groups; it is proved for $S_n$-fields of degree $n\leq5$, while the general case remains open.