Minimality conjecture for weakly maximal subgroups of branch groups

Let $G$ be a branch group and let $W\leq G$ be a weakly maximal subgroup. Write $\NR(W)$ for the non-rigidity tree of $W$, and let $\partial\NR(W)$ denote its boundary. Minimality conjecture. The action

$$W\curvearrowright\partial\NR(W)$$

is minimal. Generalized parabolic subgroups are known to have this property, and the conjecture asks whether it holds for every weakly maximal subgroup of a branch group.