The finite-index criterion for arboreal Galois groups of quadratic rational maps

Let $K$ be the field from which the rational map is defined, let $\phi\in K(x)$ be a rational function of degree $2$, and let $\mathcal{T}$ be the complete rooted $2$-ary tree attached to $\phi$. Let $G_{\phi}$ be the image of the associated arboreal Galois representation. A critical point of $\phi$ is a point $\gamma\in\mathbb P^1$ such that $\phi'(\gamma)=0$, and $\phi$ is post-critically finite if the orbit of every critical point under $\phi$ is finite. Finite-index criterion. The index of $G_{\phi}$ in $\operatorname{Aut}(\mathcal{T})$ is finite if and only if one of the following holds: (1) $\phi$ is post-critically finite; (2) the two critical points $\gamma_1$ and $\gamma_2$ satisfy $\phi^{(r+1)}(\gamma_1)=\phi^{(r+1)}(\gamma_2)$ for some $r\geq 1$; (3) $0$ is periodic under $\phi$; or (4) there is a non-trivial Möbius transformation $m$ fixing $0$ such that $\phi\circ m=m\circ\phi$. This gives a proposed classification of when the arboreal Galois image has finite index; the supplied text does not state whether the criterion has been proved or remains open.