The quadratic-polynomial arboreal Galois rigidity conjecture

For $a\in\mathbb{Z}$, let $\phi_a(x)=x^2+a$, let $T_n$ denote the rooted binary tree of height $n$, let $T_\infty$ denote the infinite rooted binary tree, and let $G_n(\phi_a)$ and $G_\infty(\phi_a)$ be the corresponding finite-level and infinite-level arboreal Galois groups. Quadratic arboreal rigidity conjecture. For every $a\in\mathbb{Z}$,

$$G_3(\phi_a)\cong\operatorname{Aut}(T_3)\quad\Longrightarrow\quad G_\infty(\phi_a)\cong\operatorname{Aut}(T_\infty).$$

In particular, if $a\neq 3$, then

$$G_2(\phi_a)\cong\operatorname{Aut}(T_2)\quad\Longrightarrow\quad G_\infty(\phi_a)\cong\operatorname{Aut}(T_\infty).$$

This conjecture would make the infinite arboreal Galois group decidable from a small iterate for the family $x^2+a$; the supplied text gives no proof or resolution status.