The abelian eighth-power subgroup conjecture for descendants of the candidate groups

Let $K=\mathbb{Q}(\sqrt{-5460})$, let $G_K$ be the Galois group of its maximal unramified $2$-extension, and let the $128$ candidate groups for $Q_3(G_K)$ be the groups identified in the preceding conjecture. For a finite or pro-$2$ group $G$, let $G^8$ denote the subgroup generated, respectively closedly generated, by all eighth powers.

Eighth-power subgroup conjecture. If $G$ is any descendant of one of the $128$ candidates for $Q_3(G_K)$, then $G^8$ has index at most $2^{40}$ and is abelian.

This observation is presented as convincing computational evidence that $G_K$ is finite, although the statement itself concerns all descendants of the candidate groups and does not by itself establish finiteness of $G_K$.