Arithmetic degree conjecture for dominant rational self-maps of varieties

Let $X$ be a nonsingular irreducible algebraic variety defined over $\overline{\mathbb{Q}}$, and let $\varphi:X\dashrightarrow X$ be a dominant rational map defined over $\overline{\mathbb{Q}}$. Choose a height function $h_X$ associated to an ample divisor, and let $X(\overline{\mathbb{Q}})_\varphi$ be the set of points whose forward orbits avoid the indeterminacy locus. Define

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

Let $\delta_\varphi=\lim_{n\to\infty}\rho((\varphi^n)^*)^{1/n}$ be the first dynamical degree, where $\rho$ denotes spectral radius on $\operatorname{NS}(X)_\mathbb{Q}$. Variety arithmetic degree conjecture. The set

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

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

This extends the projective-space arithmetic degree conjecture to arbitrary nonsingular irreducible varieties. The paper presents it as a generalization; no general proof or resolution is given.