Equality of the $K$- and $0\bb{C}_p$-Julia sets for rational maps

Let $p\ge 2$ be prime, let $K$ be a finite extension of $\mathbb{Q}_p$, and let $\mathbb{C}_p$ be the completion of an algebraic closure of $\mathbb{Q}_p$. For a rational map $\phi\in K(z)$, write $J_K(\phi)$ and $J_{\mathbb{C}_p}(\phi)$ for its Julia sets on $\mathbb{P}^1_K$ and $\mathbb{P}^1_{\mathbb{C}_p}$, respectively.

Julia-set equality conjecture. For every rational map $\phi\in K(z)$,

$$J_{\mathbb{C}_p}(\phi)\cap\mathbb{P}^1_K=J_K(\phi).$$

The conjecture asks whether Julia points defined over $K$ are exactly the points of the $\mathbb{C}_p$-Julia set that lie in $\mathbb{P}^1_K$. An affirmative answer would establish consistency between the Julia sets over a finite extension of $\mathbb{Q}_p$ and over $\mathbb{C}_p$; the source provides no resolution.