Arboreal Sen Conjecture for PCF maps

Let $K$ be a finite extension of $\mathbb{Q}_p$ and let $\phi \in K(x)$ be a PCF rational function of degree $d \geq 2$. Let $b \in \mathbb{P}^1(K)$ be not exceptional for $\phi$.

Arboreal Sen Conjecture for PCF maps. If $p \mid d$, then the iterated extension

$$K(\phi^{-\infty}(b))$$

is a branch-APF extension of $K$.

This conjecture proposes a dynamical analogue of Sen's theorem for PCF maps. The paper proves branch-APF results for several families of rational functions and restricted basepoints, while the assertion for every PCF rational map of degree divisible by $p$ remains open.