Bhargava's asymptotic conjecture for degree-$d$ number fields with Galois group $S_d$

Let $N_d(X)$ denote the number of number fields of degree $d$ having discriminant with absolute value at most $X$. Let $q(n,k)$ denote the number of partitions of $n$ into at most $k$ parts, and let $r_2(S_d)$ denote the number of elements of order either $1$ or $2$ in $S_d$. Bhargava's conjecture. The asymptotic density satisfies

$$\lim_{X \to \infty} \frac{N_d(X)}{X} = \frac{r_2(S_d)}{2d!} \prod_{p \text{ prime}} \left( \frac{\sum_{k=0}^n q(k,n-k)-q(k-1,n-k+1)}{p^k} \right).$$

This conjecture predicts the asymptotic growth of degree-$d$ number fields with Galois group $S_d$ as a function of the discriminant. The supplied passage does not state whether it has been resolved.