Kawaguchi–Silverman conjecture for surjective endomorphisms

Let $f:X\to X$ be a surjective endomorphism of a projective variety $X$ defined over $\overline{\mathbb{Q}}$. For $x\in X(\overline{\mathbb{Q}})$, write

$$O_f(x):=\{f^n(x)\mid n\geq 0\}.$$

Assume that $O_f(x)$ is Zariski dense in $X$.

Kawaguchi–Silverman conjecture. The arithmetic degree at $x$ equals the dynamical degree of $f$:

$$\alpha_f(x)=\delta_f.$$

This conjecture compares arithmetic complexity along a forward orbit with the geometric complexity of the dynamical system. It is stated here for surjective endomorphisms, while the original formulation concerns dominant self-maps.