Local Pre-Malle conjecture for non-reduced discriminants

Let $G\leq S_n$ be a transitive permutation group, and define

$$e_0=\gcd\{\operatorname{ind}(g):g\in G\setminus\{1\}\}.$$

Here $\operatorname{ind}(g)$ is the index of the permutation $g$. A discriminant is the discriminant of a $G$-extension of $\mathbb{Q}$.

Local Pre-Malle conjecture. There is a finite set of primes $S_0$ depending on $G$ such that, for every prime $p\notin S_0$ and every integer $k\not\equiv 0\pmod p$, some $G$-extension of $\mathbb{Q}$ has discriminant $\Delta\equiv k^{e_0}\pmod p$. The supplied text proposes this as the non-reduced-discriminant analogue of the preceding local conjecture; its status is not resolved there.