Property H conjecture for subdegree-finite primitive permutation groups

Let $G$ be a subdegree-finite primitive permutation group with more than one end. Recall that $G$ has property H if, in its faithful action on its structure tree $T$, the pointwise stabiliser of every half-tree is nontrivial, where a half-tree is a component of $T\setminus e$ for an edge $e$ whose two complementary components are infinite. Property H conjecture. $G$ has property H. If true, this would identify the unique minimal closed normal subgroup $B$ of $G$ with $G^{++}$, the closure of the subgroup generated by the pointwise stabilisers of half-trees, and would imply that $B$ is topologically simple.