Counting conjecture for $S_{d+1}$-fields with local conditions

Fix $d+1\geq3$ and a signature $(r_1,r_2)$. Let $L(X)^{r_2}$ be the set of $S_{d+1}$-fields $K$ with $X/2<|D_K|<X$, and let $L(X;\mathcal S)^{r_2}$ be the subset satisfying finitely many local conditions $\mathcal S$ at primes in a finite set $S$. Write $A(r_2)$ for the expected density constant and $\lvert\mathcal S\rvert$ for the product of the local densities. Counting conjecture. There are positive constants $\delta<1$ and $\kappa$ such that

$$|L(X)^{r_2}|=A(r_2)X+O(X^\delta),$$ $$|L(X;\mathcal S)^{r_2}|=|\mathcal S|A(r_2)X+O\left(\left(\prod_{p\in S}p\right)^\kappa X^\delta\right),$$

with the implied constant uniformly bounded for $p$ and the local conditions at $p$. This conjectural uniform counting estimate is used to control local conditions in the construction of fields with extreme residues; the supplied text does not state that it is proved in full generality.