The three-stage surjectivity conjecture for quadratic arboreal representations

Let $\phi_c(x)=x^2+c$ with $c\in\mathbb{Z}$. Write $G_n(\phi_c)$ for the Galois group at the $n$th stage of the arboreal representation and $G_\infty(\phi_c)$ for its full image, viewed as subgroups of the automorphism groups $\operatorname{Aut}(T_{2,n})$ and $\operatorname{Aut}(T_2)$ of the rooted binary trees of height $n$ and infinite height, respectively. Three-stage surjectivity conjecture. If

$$G_3(\phi_c)=\operatorname{Aut}(T_{2,3}),$$

then

$$G_\infty(\phi_c)=\operatorname{Aut}(T_2).$$

This conjecture predicts that surjectivity at the third iterate already forces surjectivity of the full arboreal representation for every integer parameter in the family $x^2+c$. The paper gives evidence through explicit low-level results and searches, but the asserted implication is not proved in the supplied text.