Arboreal Galois groups for cubic polynomials with colliding critical points

Let $K$ be a field, and let $f\in K(z)$ be a rational function of degree $d\ge 2$. The Galois group of the field extension generated by the preimages of $x_0\in K$ under all iterates of $f$ naturally embeds in the automorphism group of an infinite $d$-ary rooted tree. In some cases the Galois group can be the full automorphism group of the tree, but in other cases it is known to have infinite index. In this paper, we consider a previously unstudied such case: that $f$ is a polynomial of degree $d=3$, and the two finite critical points of $f$ collide at the $\mathrm{\ell }$-th iteration, for some $\mathrm{\ell }\ge 2$. We describe an explicit subgroup $Q_{\mathrm{\ell },\mathrm{\infty }}$ of automorphisms of the $3$-ary tree in which the resulting Galois group must always embed, and we present sufficient conditions for this embedding to be an isomorphism.