Malle–Bhargava heuristics for prescribed local conditions

Let $k$ be a number field, let $n$ be a positive integer, and let $\Sigma=(\Sigma_\mathfrak{p})_\mathfrak{p}$ be an acceptable degree $n$ collection of local conditions on $k$. Write $\mathbf{Ext}_{k,n}$ for the isomorphism classes of degree $n$ extensions $K/k$ whose Galois closure has group $S_n$, and define

$$\mathbf{Ext}_{k,n,\leq X}=\{K\in\mathbf{Ext}_{k,n}:\operatorname{Nm}(\operatorname{disc}(K/k))\leq X\}.$$

For such a collection, let

$$N_{k,n}(X;\Sigma)=\#\{K\in\mathbf{Ext}_{k,n,\leq X}:K\text{ satisfies }\Sigma\}.$$

Malle–Bhargava heuristics. One expects

$$\lim_{X\to\infty}\frac{N_{k,n}(X;\Sigma)}{X}=\frac{1}{2}\cdot\operatorname{Res}_{s=1}\bigl(\zeta_k(s)\bigr)\cdot\prod_{\mathfrak{p}\in\Pi_k}m(\Sigma_\mathfrak{p}).$$

This predicts an asymptotic count of degree $n$ extensions with Galois closure group $S_n$, bounded by discriminant norm and subject to prescribed acceptable local conditions. The supplied text does not establish the assertion or provide evidence resolving its status.