The positive-dimensional subgroup conjecture in automorphism groups of rooted trees

Let $\Gamma(p)$ be the profinite automorphism group of the rooted $p$-ary tree, and let positive-dimensional mean having positive Hausdorff dimension as a closed subgroup of $\Gamma(p)$. A nonabelian free pro-$p$ subgroup is a closed subgroup isomorphic to a free pro-$p$ group of rank at least two. The positive-dimensional subgroup conjecture. If $G\subseteq \Gamma(p)$ is a positive-dimensional closed subgroup, then $G$ contains a nonabelian free pro-$p$ subgroup. The conjecture strengthens the free-subgroup question for branch groups; the source presents it as an open conjecture and notes its relation to identities and linearity.