Boston's branch-group conjecture for just-infinite pro-$p$ groups

Let $p$ be a prime, let $T_p$ be the rooted $p$-ary tree, and let $W_p$ be its full automorphism group. A pro-$p$ group is just-infinite if all its non-trivial closed normal subgroups are open, and it is branch if it acts level-transitively on $T_p$ with finite-index rigid stabilizers at every level. An embedding of a pro-$p$ group into $W_p$ has positive Hausdorff dimension when its image has positive Hausdorff dimension in the profinite group $W_p$. Boston's conjecture. A just-infinite pro-$p$ group is branch if and only if it admits an embedding into $W_p$ with positive Hausdorff dimension. The conjecture is a geometric group-theoretic characterization of branch just-infinite pro-$p$ groups and was motivated by their role in the study of maximal unramified pro-$p$ extensions and the Fontaine–Mazur conjecture. The paper disproves this conjecture.