Matsuzawa–Matsumoto–Sano–Zhang conjecture on bounded-degree points with non-maximal arithmetic degree

Let $X$ be a projective variety over a number field $K$, let $f:X\to X$ be a surjective morphism, and let $d>0$. Define

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

Matsuzawa–Matsumoto–Sano–Zhang conjecture. The set $Z_f(d)$ is not Zariski dense in $X$. This concerns the distribution of algebraic points of bounded degree whose arithmetic degree is strictly smaller than the first dynamical degree; the source presents it as a conjecture attributed to Matsuzawa, Matsumoto, Sano, and Zhang, and does not provide evidence of resolution.