Dynamical Lang height conjecture

Let $K/\mathbb{Q}$ be a number field, let $\mathcal{M}_d^N$ be the moduli space of degree-$d$ endomorphisms of $\mathbb{P}^N$, and let $h_{\mathcal{M}}$ be an ample height on $\mathcal{M}_d^N(\overline{\mathbb{Q}})$. For $f\in\mathcal{M}_d^N(K)$ and $P\in\mathbb{P}^N(K)$, let $\widehat h_f(P)$ denote the canonical height. Dynamical Lang height conjecture. There exist constants $C_1(K,N,d)>0$ and $C_2(K,N,d)$ such that, whenever the $f$-orbit of $P$ is Zariski dense,

$$\widehat h_f(P)\geq C_1h_{\mathcal{M}}(f)-C_2.$$

This predicts that points with dense orbit have canonical height bounded below in terms of the moduli height of the dynamical system. The general conjecture remains open.