The centralizer index conjecture for rational maps with automorphisms

Let $K$ be a field, let $\phi\in K(x)$, and let $T_\infty$ be its pre-image tree rooted at $0$. Let $A(\phi)$ be the subgroup of $\operatorname{Aut}(T_\infty)$ generated by the actions of all nontrivial Möbius transformations $m\in\operatorname{PGL}_2(K)$ satisfying $m^{-1}\circ\phi\circ m=\phi$ and $m(0)=0$. Let $C(\phi)$ be the centralizer of $A(\phi)$ in $\operatorname{Aut}(T_\infty)$. Centralizer index conjecture. If $A(\phi)$ is nontrivial, then

$$[\operatorname{Aut}(T_\infty):C(\phi)]=\infty.$$

The centralizer contains the arboreal Galois group because the Galois action commutes with the $K$-defined Möbius symmetries. The source gives no resolution of this conjecture.