Full generic arboreal Galois group conjecture for rational functions

Let $k$ be a number field, and let $\phi(z)\in k(z)$ be a rational function of degree $d\geq 2$, defined over $k$. Let $L=k(t)$ be the function field, and define the generic arboreal Galois group with root point $t$ by

$$G(\phi,k):=\operatorname{Gal}(L_{\infty}/L),\qquad L_{\infty}:=\bigcup_{n\geq 0}L\big(\phi^{-n}(t)\big).$$

For a finite extension $K/k$ and $x_0\in\mathbb{P}^1(K)$, let $K_{\infty}$ be the field generated by the iterated preimages of $x_0$ under $\phi$, and let $G_{\infty}:=\operatorname{Gal}(K_{\infty}/K)$. Full generic arboreal group conjecture. For every finite extension $K/k$ and every $x_0\in\mathbb{P}^1(K)$, the group $G_{\infty}$ is isomorphic to a subgroup of $G(\phi,k)$; moreover, one can choose $K$ and $x_0$ so that $G_{\infty}$ is the full group $G(\phi,k)$. This broader conjecture predicts that the generic function-field arboreal group is realized by a suitable specialization, while every specialization gives a subgroup of it.