Matsuzawa–Meng–Zhang–Sano conjecture on bounded-degree points of maximal arithmetic degree

Let $X$ be a projective variety over a number field $K$, let $f\colon X \longrightarrow X$ be a surjective morphism, and let $d>0$ be a positive integer. Define

$$Z_f(d):=\{x\in X(\overline{K})\mid [K(x):K]\leq d,\ \alpha_f(x)<\delta_f\}.$$

Matsuzawa–Meng–Zhang–Sano conjecture. The set $Z_f(d)$ is not Zariski dense in $X$. This conjecture predicts that, among points of bounded degree, points whose arithmetic degree is strictly below the dynamical degree cannot be Zariski dense. The source attributes the conjecture to Matsuzawa, Meng, Zhang, and the second author; the supplied text gives no resolution status.