Göksel's Markov group containment conjecture

Let $f$ be the post-critically finite quadratic polynomial under consideration, and let $G_n(f)$ and $M_n(f)$ denote, respectively, its level-$n$ Galois group and even Markov group. Göksel's Markov group containment conjecture. For all $n\geq 1$, $G_n(f)$ is isomorphic to a subgroup of $M_n(f)$. This conjecture refines the Boston–Jones Markov heuristic and predicts that the even Markov groups contain the corresponding Galois groups at every level; the supplied context gives no resolution.