Hyperbolic polynomial branch and extension conjecture

Let $K$) be a field and let $f\bin K[z]$ be a hyperbolic polynomial in the orbifold sense, with $L_\infty\ne K$. Write $G_\infty(K,f,t)$ and $G_\infty(K^\mathrm{sep},f,t)$ for the arithmetic and geometric iterated Galois groups, respectively. The extension $L_\infty/K$ is the associated extension in the arboreal Galois representation.

Hyperbolic polynomial branch and extension conjecture. The following are equivalent:

1. Both $G_\infty(K,f,t)$ and $G_\infty(K^\mathrm{sep},f,t)$ are branch.
2. The extension $L_\infty/K$ is finite.
3. None of the critical points of $f$ in $K$ is periodic.

This conjecture relates branching of arithmetic and geometric iterated Galois groups to finiteness of the extension and the absence of periodic critical points. The source presents it as a conjecture after proving that branching implies finiteness, but gives no resolution of the equivalence.