Bellon–Viallet integrality conjecture for dynamical degrees

Let $f:\mathbb{P}^N\dashrightarrow\mathbb{P}^N$ be a dominant rational map, and let

$$\delta(f):=\lim_{n\to\infty}\bigl(\deg(f^n)\bigr)^{1/n}$$

be its dynamical degree. Bellon–Viallet conjecture. The dynamical degree is an algebraic integer:

$$\delta(f)\in\overline{\mathbb{Z}}.$$

The conjecture holds for several classes of varieties and maps, including birational maps of $\mathbb{P}^2$, but the paper notes counterexamples in the stated generality: dominant rational maps of $\mathbb{P}^2$ and birational maps of $\mathbb{P}^N$ for $N\geq3$ can have transcendental dynamical degree.