The family conjecture for quadratic rational maps

Let $\phi(x)=b(x^2+1)/x$ with $b\in\mathbb{Q}$ and $b\notin\{0,\pm 1/2\}$. 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 $0$. Family conjecture. One has

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

This extends the finite-index results known for several congruence classes and for many small values of $b$. The source attributes the conjecture to the analysis of Garton and Rafe Jones and gives no resolution.