Positive canonical height conjecture for dense orbits

Let $\varphi:\mathbb{P}^N\dashrightarrow\mathbb{P}^N$ be a dominant rational map defined over $\overline{\mathbb{Q}}$, with dynamical degree $\delta_\varphi>1$. Let $\ell_\varphi$ be the polynomial correction exponent from the degree-growth conjecture, and for $P\in\mathbb{P}^N(\overline{\mathbb{Q}})_\varphi$ define

$$\widehat h_\varphi(P)=\limsup_{n\to\infty}\frac{h(\varphi^n(P))}{n^{\ell_\varphi}\delta_\varphi^n}.$$

Here $\mathbb{P}^N(\overline{\mathbb{Q}})_\varphi$ consists of points whose forward orbits avoid the indeterminacy locus. Positive canonical height conjecture. If the orbit of $P$ is Zariski dense in $\mathbb{P}^N$, then

$$\widehat h_\varphi(P)>0.$$

For any map with $\delta_\varphi>1$ or $\ell_\varphi>0$, finite orbits have canonical height zero, but the converse can fail on invariant subvarieties with lower degree growth. The conjecture proposes the converse for Zariski-dense orbits; it is proved in the paper for certain monomial maps.