The finitely generated full-dimension conjecture for closed subgroups

Let $\Gamma(p)$ be the profinite automorphism group of the rooted $p$-ary tree, let $G$ be a closed subgroup, and let Hausdorff dimension be computed in $\Gamma(p)$. A subgroup is topologically finitely generated if it has a finite subset whose closed subgroup generated by that subset is the subgroup itself. The finitely generated full-dimension conjecture. The group $G$ contains a topologically finitely generated subgroup having the same Hausdorff dimension as $G$. This is one of two conjectures proposed in increasing strength concerning finitely generated subgroups of $\Gamma(p)$; the source gives no resolution.