Rigid stabilizers and local prosolubility for boundary-transitive actions on trees

Let $G$ be a group acting $2$-transitively on the boundary of a locally finite tree, and exclude the situation (which is a genuine exception) where $G$ has both $\mathrm{P}\Gamma\mathrm{L}_3(4)$ and $\mathrm{P}\Gamma\mathrm{L}_3(5)$ as local actions. We show that for each half-tree $T_a$, the local action of the rigid stabilizer of $T_a$ at the root of $T_a$ contains the soluble residual of the point stabilizer of the local action of $G$. In particular, $G$ is locally prosoluble if and only if its local actions have soluble point stabilizers; if $G$ is not locally prosoluble, then it has micro-supported action on the boundary. We also prove some strong restrictions on the local actions of end stabilizers in $G$. These results are partly inspired by Radu's classification of groups acting boundary-$2$-transitively on trees with local action containing the alternating group, and partly based on the author's recent classification of finite permutation groups that preserve an equivalence relation, act faithfully on blocks and act transitively on pairs of points from different blocks.