Jones's index conjecture for quadratic rational maps

Let $K$ be a global field, let $f\in K(x)$ have degree two, and let $T_\infty$ be the infinite rooted binary tree of preimages of a root $\alpha$ under $f$. Write $G_\infty(f,\alpha)$ for the image of the associated arboreal Galois representation. Jones's conjecture. One has

$$[\operatorname{Aut}(T_\infty):G_\infty(f,\alpha)]=\infty$$

if and only if at least one of the following holds: $f$ is postcritically finite; the two critical points $\gamma_1$ and $\gamma_2$ satisfy $f^{r+1}(\gamma_1)=f^{r+1}(\gamma_2)$ for some $r\geq 1$; the root $\alpha$ is periodic under $f$; or a nontrivial Möbius transformation commuting with $f$ fixes $\alpha$. This gives criteria for when the arboreal image has infinite index in the full automorphism group; the supplied text does not indicate whether the conjecture has been resolved.