Bhargava's Chebotarev splitting conjecture for $S_n$-number fields

Let $n$ be a positive integer and let $B>0$. Let $\mathcal{G}_{n}(B;p)$ be the collection of degree-$n$ number fields $K$ such that $(p)$ is unramified in $K$, $|D_K|\leq B$, and the normal closure of $K/\mathbb{Q}$ has Galois group $S_n$. Let $T_n^{\ast}$ denote the set of partitions of $n$ corresponding to conjugacy classes of $S_n$, let $\mu\in T_n^{\ast}$, and let $C_\mu$ be the associated conjugacy class. Define the Chebotarev density distribution by

$$\nu_n(C_\mu):=\frac{|C_\mu|}{|S_n|}.$$

Bhargava's Chebotarev splitting conjecture. For every prime $p$, positive integer $n$, and $\mu\in T_n^{\ast}$,

$$\lim_{B\rightarrow\infty}\frac{\#\{K\in\mathcal{G}_n(B;p)\mid p\text{ has Artin symbol in }C_\mu\}}{\#\{K\in\mathcal{G}_n(B;p)\}}=\nu_n(C_\mu).$$

The conjecture predicts that, after restricting to fields unramified at $p$, Artin symbols are distributed according to the Chebotarev densities of conjugacy classes in $S_n$. The supplied text does not report a proof or disproof, so the conjecture is left open.