The general symmetry conjecture for quadratic rational maps

Let $K$ be a global field of characteristic $0$ or greater than $2$, and let $\phi(x)\in K(x)$ have degree $2$. Assume that $\phi$ is not post-critically finite and that $0$ is not periodic under $\phi$. Let $T_\infty$ be the pre-image tree rooted at $0$, let $G_\infty(\phi)$ be its arboreal Galois group, and let $C(\phi)$ be the centralizer of the subgroup generated by Möbius symmetries commuting with $\phi$ and fixing $0$. General symmetry conjecture. If $\phi$ commutes with a nontrivial Möbius transformation fixing $0$, then

$$[C(\phi):G_\infty(\phi)]<\infty.$$

Every degree-two rational map commuting with a nontrivial Möbius transformation is conjugate to the displayed family, so this conjecture generalizes the corresponding case over $\mathbb{Q}$. The source presents it as equivalent to the family conjecture over $\mathbb{Q}$ and gives no resolution.