The generalized semisimplicity conjecture for polarized endomorphisms

Let $X$ be a smooth projective variety of dimension $n$ over $\mathbf{K}$, and let $f$ be a polarized self-morphism as in the generalized Weil's Riemann hypothesis: $f^*H_X\sim_{\operatorname{rat}}qH_X$ for an ample divisor $H_X$ and some $q\in\mathbf{N}_{\geq 2}$. Generalized semisimplicity conjecture. For every $0\leq i\leq 2n$, the pullback action

$$f^*|_{H^i(X)}$$

on the $\ell$-adic étale cohomology group $H^i(X)$ is diagonalizable over $\overline{\mathbf{Q}}_\ell$. This is presented as a consequence of the standard conjectures and is open in the stated generality.