Wood's moment conjecture for admissible pairs of 2-groups

Let $K$ be a quadratic field with discriminant $d$, let $(G',G)$ be an admissible pair of $2$-groups in Wood's sense, and let $f(d)$ be the number of unramified $(G',G)$-extensions of $K$. Let $\omega(d)$ denote the number of distinct prime divisors of $d$, and let $\mathbf{E}$ denote the corresponding average over quadratic fields.

Wood's moment conjecture. There exists a number $0<a<1$ such that, for every positive integer $k$,

$$\mathbf{E}\left(\left(a^{\omega(d)}f(d)\right)^{k}\right)<\infty,$$

and these moments determine a distribution. In the case $(G',G)=(D_{4}\times C_{2},D_{4})$, one has $a=1/4$.

This conjecture predicts finite normalized moments and a determined limiting distribution for counts of unramified extensions. The paper proves moment asymptotics for the pair $(H_{8}\rtimes C_{2},H_{8})$, but the general assertion and the specified $D_{4}\times C_{2}$ case remain open.