Bhargava's ramification-density conjecture for $S_n$-number fields

Let $n$ be a positive integer and let $B>0$. Let $\mathcal{G}_{n}(B)$ be the collection of $S_n$-number fields $K$ with $|D_K|\leq B$. For a prime $p$, let $\mathcal{G}_{n}(B;p)$ be the collection of degree-$n$ number fields satisfying: $(p)$ is unramified in $K$, $|D_K|\leq B$, and the normal closure of $K/\mathbb{Q}$ has Galois group $S_n$. Let $q(k,n)$ denote the number of partitions of $k$ into at most $n$ parts, with $q(0,n)=1$ for $n\geq 1$, and define

$$\rho_n(p):=\frac{\sum_{k=1}^{n-1}q(k,n-k)p^{n-1-k}}{\sum_{k=0}^{n-1}q(k,n-k)p^{n-1-k}}.$$

Bhargava's ramification-density conjecture. For every prime $p$ and positive integer $n$,

$$\lim_{B\rightarrow\infty}\frac{\#\{K\in\mathcal{G}_n(B;p)\}}{\#\{K\in\mathcal{G}_n(B)\}}=1-\rho_n(p).$$

Here $\rho_n(p)$ is the predicted probability of ramification at $p$. Bhargava proves the conjecture for $n\leq 5$; the assertion remains conjectural for general $n$.