Existence of an algebraically stable ramified model for maps with large dynamical degree

Let $\f: \mathbb{P}^2 \dashrightarrow \mathbb{P}^2$ be a rational map of topological degree $e$, and let $\lambda_1=\lim_{n\to\infty}\operatorname{deg}(\f^n)^{1/n}$ be its first dynamical degree. A rational map is algebraically stable (AS) if $\bigcup_{k\geq 0}\f^{-k}I(\f)$ is a countable union of points, where $I(\f)$ is its indeterminacy set. The ramified-model conjecture. If

$$e^2<\lambda_1,$$

then there exist a birational model $\overline{X}$ of $\mathbb{P}^2$ and a ramified covering $h:X\to\overline{X}$ such that $\f$ lifts to $X$ and becomes AS. The conjecture proposes an algebraically stable realization after passing to a ramified cover, in a regime where the dynamical degree dominates the square of the topological degree; the supplied text gives no resolution evidence.