Local Pre-Malle conjecture for reduced discriminants

Let $G$ be a finite group. A reduced discriminant is the reduced 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 reduced discriminant $\delta\equiv k\pmod p$. Moreover, if $p\in\mathcal{P}(G)$, the same assertion holds for the zero residue class. Equivalently, apart from the trivial obstructions, only finitely many local obstructions occur for reduced discriminants of $G$-extensions. The supplied text presents this as a conjectural strengthening related to the positive-density conjecture.