The discriminant multiplicity conjecture for number-field extensions

Let $F$ be a number field and let $G\leq S_n$ be a transitive permutation group. For an extension $E/F$ whose Galois closure has permutation group $G$, define

$$a_D:=\#\{E/F\mid \operatorname{Gal}(E/F)=G,\ \operatorname{Disc}(E/F)=D\}.$$

Discriminant multiplicity conjecture. For every $D\in\mathbb{N}$, one has

$$a_D=O_{\epsilon,F,n}(D^{\epsilon})$$

for all $\epsilon>0$.

The conjecture asserts that the number of extensions with a fixed discriminant is smaller than every fixed positive power of that discriminant. The source relates it to Malle's conjecture and the $\ell$-torsion conjecture, proving the corresponding results for nilpotent groups; the general statement remains open.