The generalized Weil's Riemann hypothesis for polarized endomorphisms

Let $X$ be a smooth projective variety of dimension $n$ over $\mathbf{K}$, and let $f$ be a self-morphism of $X$. Suppose that $f$ is polarized, meaning that

$$f^*H_X\sim_{\operatorname{rat}}qH_X$$

for an ample divisor $H_X$ on $X$ and a positive integer $q\in\mathbf{N}_{\geq 2}$. A $q$-Weil number of weight $i$ is an algebraic number $\alpha$ satisfying $|\sigma(\alpha)|=q^{i/2}$ for every embedding $\sigma\colon\overline{\mathbf{Q}}\hookrightarrow\mathbf{C}$. Generalized Weil's Riemann hypothesis. For every $0\leq i\leq 2n$, the eigenvalues of $f^*|_{H^i(X)}$ are $q$-Weil numbers of weight $i$. This is a positive-characteristic analogue of a result for compact Kähler manifolds and follows from the standard conjectures; it is therefore open in the generality stated.