The liminf Height Gap Conjecture

Let $X$ be a quasi-projective variety, let $\Phi\colon X\dashrightarrow X$ be a rational self-map, and let $f\colon X\dashrightarrow \mathbb{P}^1$ be a non-constant rational function, all defined over $\overline{\mathbb{Q}}$. Let $X_{\Phi,f}(\overline{\mathbb{Q}})$ denote the points whose forward $\Phi$-orbits avoid the indeterminacy loci of both $\Phi$ and $f$, and let ${\mathcal O}_\Phi(x)$ denote the forward orbit of $x$. Let $h$ be the absolute logarithmic Weil height. The liminf Height Gap Conjecture. If $X$ is irreducible and ${\mathcal O}_\Phi(x)$ is Zariski dense in $X$, then

$$\liminf_{n\to\infty} \frac{h(f(\Phi^n(x)))}{\log n}>0.$$

This conjecture was introduced to imply the Dynamical Mordell--Lang Conjecture. The source does not state a proof or disproof of this liminf conjecture; its status is therefore open, although the paper proves the liminf statement away from a set of density zero.