The discriminant multiplicity conjecture

For each integer $n\geq 2$, let $D_K$ denote the absolute discriminant of a degree-$n$ number field $K/\mathbb{Q}$. The discriminant multiplicity conjecture. For every $\epsilon>0$, there exists a constant $C_{n,\epsilon}$ such that, for every integer $D\geq 1$, at most $C_{n,\epsilon}D^{\epsilon}$ degree-$n$ fields $K/\mathbb{Q}$ have $D_K=D$. This is a folk conjecture related to counting number fields with fixed discriminant. It is known that for $n\geq 6$ no improvement over the trivial available bound is known for the corresponding discriminant-multiplicity property.