Moment and asymptotic conjecture for admissible finite 2-groups

Let $G$ be a finite $2$-group and let $H\leq G$ have index $2$, with $(G,H)$ admissible. Let $f(K)$ be the associated unramified-extension counting function, let $c$ be the corresponding constant, let $\omega(D_K)$ denote the number of prime divisors of the discriminant, and let $\boldsymbol{E}_{k}^{\pm}$ denote averages over quadratic fields of positive or negative discriminant. Moment and asymptotic conjecture. For every positive integer $k$, one has

$$0<\boldsymbol{E}_{k}^{\pm}\left(\frac{f}{c^{\omega(D_K)}}\right)<\infty$$

and these moments determine a distribution on the values of $f$. Moreover, there is a constant $C(G,H)$ such that

$$\sum_{K,0<\pm D_K<X}f^{k}(K)\sim C(G,H)X(\log X)^{c^{k}-1}.$$

The conjecture extends known moment calculations for particular groups and predicts the logarithmic asymptotics of unramified extension counts for admissible finite $2$-groups.