Positive lower ample canonical height for Zariski-dense orbits

Let $X$ be a smooth projective variety over $\overline{\mathbb Q}$, let $f$ be an endomorphism of $X$ with $\delta_f>1$, and for $x\in X(\overline{\mathbb Q})$ write

$$O_f(x)=\{x,f(x),f^2(x),\ldots\}.$$

Positive lower ample canonical height conjecture. If $O_f(x)$ is dense in the Zariski topology, then $\underline h_f(x)>0$. The source describes this as a weaker conjecture and as a generalization of a conjecture of Kawaguchi and Silverman restricted to endomorphisms. Its general status is open.