Local-to-global conjugacy conjecture for Markov groups

Let $\mathcal{T}_n$ be the rooted ternary tree of height $n$, and let $M$ be the group arising from Sections 3 and 4. For subgroups $H,G\leq\operatorname{Aut}(\mathcal{T}_n)$, say that $H$ is locally conjugated into $G$ if, for every $h\in H$, there exists $k$ in the kernel of the restriction map

$$\operatorname{Aut}(\mathcal{T}_n)\to\operatorname{Aut}(\mathcal{T}_{n-1})$$

such that $h^k\in G$. Say that $H$ is globally conjugated into $G$ if there exists one such $k$ with $H^k\subseteq G$.

Local-to-global conjugacy conjecture. If $H\leq\operatorname{Aut}(\mathcal{T}_n)$ is locally conjugated into $M$, then $H$ is globally conjugated into $M$.

This is a group-theoretic conjecture intended to reduce questions about the Markov model to conjugacy inside automorphism groups of rooted trees. The source recalls it in the setting of the paper and does not state that it has been resolved.