Göksel's transitive-subgroup conjugacy conjecture

Let $n\geq 0$, let $T_n$ denote the first $n$ levels of the rooted binary tree, and let $H\leq \operatorname{Aut}(T_n)$ be a group that contains an element acting transitively on $\{1,\ldots,2^n\}$. Let $\mathcal{P}(H,G)$ denote the elementwise conjugacy property discussed in the paper. Göksel's transitive-subgroup conjugacy conjecture. Then $\mathcal{P}(H,G)$ holds for any $G\leq \operatorname{Aut}(T_n)$. This is a group-theoretic question motivated by the attempt to prove that Galois groups are conjugate to subgroups of even Markov groups; the supplied context gives no resolution, while related generic cases are stated to be known.