The limsup 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 limsup Height Gap Conjecture. For any $x\in X_{\Phi,f}(\overline{\mathbb{Q}})$, either $f({\mathcal O}_\Phi(x))$ is finite, or

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

This conjecture was introduced to imply the Dynamical Mordell--Lang Conjecture and results on the growth of coefficients of $D$-finite power series. It has been proved in the paper: the source states that its first main result is a proof of the conjecture, generalizing the previously known case where $\Phi$ and $f$ are morphisms.