The non-discrete two-boundary-point stabiliser conjecture

Let $T$ be a locally finite tree and let $G\leq\operatorname{Aut}(T)$ be closed, non-compact, and boundary-transitive. For distinct boundary points $\xi,\xi'\in\partial T$, write $G_{\xi,\xi'}$ for their common stabiliser. Non-discrete stabiliser conjecture. The stabiliser $G_{\xi,\xi'}$ is not discrete. This is the original formulation of the conjecture; the paper relates it to the boundary-Moufang formulation below via a lemma showing that a discrete stabiliser forces the boundary-Moufang property.