Shibata's conjecture on ample canonical heights

Let $X/\overline{\mathbb{Q}}$ be a smooth projective variety, let $f:X\to X$ be a dominant morphism with $\delta(f)>1$, and define

$$\ell(f):=\inf\left\{\ell\geq0:\sup_{n\geq1}\frac{h_X(f^n(P))}{n^\ell\delta(f)^n}<\infty\right\}$$

with the infimum of the empty set defined as $\infty$. Define the lower ample canonical height by

$$\underline{\widehat h}_f(P):=\liminf_{n\to\infty}\frac{h_X(f^n(P))}{n^{\ell(f)}\delta(f)^n},$$

setting it to $0$ if $\ell(f)=\infty$. Shibata's conjecture. (a) The quantity $\ell(f)$ is a non-negative integer. (b) For every number field $K/\mathbb{Q}$ over which $X$ and $f$ are defined, the set

$$\{P\in X(K):\underline{\widehat h}_f(P)=0\}$$

is not Zariski dense in $X$. The paper uses this conjecture to imply the Kawaguchi–Silverman density conjecture for morphisms; the general assertions remain open.