Arithmetic degree conjecture for dominant rational self-maps of projective space

Let $\varphi:\mathbb{P}^N\dashrightarrow\mathbb{P}^N$ be a dominant rational map defined over $\overline{\mathbb{Q}}$. For a point $P\in\mathbb{P}^N(\overline{\mathbb{Q}})$ whose orbit avoids the indeterminacy locus, write $\mathbb{P}^N(\overline{\mathbb{Q}})_\varphi$ for the set of such points, and define

$$\alpha_\varphi(P)=\limsup_{n\to\infty}h\bigl(\varphi^n(P)\bigr)^{1/n}.$$

Let $\delta_\varphi=\lim_{n\to\infty}(\deg(\varphi^n))^{1/n}$ be the dynamical degree. Arithmetic degree conjecture. The set

$$\bigl\{\alpha_\varphi(P):P\in\mathbb{P}^N(\overline{\mathbb{Q}})_\varphi\bigr\}$$

is a finite set of algebraic integers; moreover, if the orbit of $P\in\mathbb{P}^N(\overline{\mathbb{Q}})_\varphi$ is Zariski dense in $\mathbb{P}^N$, then $\alpha_\varphi(P)=\delta_\varphi$.

This conjecture relates height growth of individual orbits to the global degree growth of the map. The paper proves it for monomial maps on projective space, while the general case remains open.