The quadratic resolvent counting conjecture

Let $F$ be a quadratic field, and let $QR_n(X,F)$ be the set of $S_n$-fields whose common quadratic resolvent is $F$ and whose absolute discriminant is bounded by $X$. Let $d_F$ denote the discriminant of $F$.

Quadratic resolvent counting conjecture. There is a constant $\beta_n$ with $0<\beta_n<1/2$ such that

$$|QR_n(X,F)|\ll\left(\frac{X}{|d_F|}\right)^{1-\beta_n},$$

where the implied constant is independent of $F$.

The conjecture is a weaker form of the expected bound for fields sharing a quadratic resolvent and is used in the paper to obtain results on the smallest prime in a conjugacy class. Its status is not resolved in the supplied text.