An inequality on polarized endomorphisms

We show that assuming the standard conjectures, for any smooth projective variety $X$ of dimension $n$ over an algebraically closed field, there is a constant $C>0$ such that for any positive rational number $r$ and for any polarized endomorphism $f$ of $X$, we have $$\| G_r \circ f \| \le C \, \mathrm{deg}(G_r \circ f),$$ where $G_r$ is a correspondence of $X$ so that for each $0\le i\le 2n$ its pullback action on the $i$-th Weil cohomology group is the multiplication-by-$r^i$ map. This inequality has been conjectured by the authors to hold in a more general setting, which - in the special case of polarized endomorphisms - confirms the validity of the analog of a well known result by Serre in the K\"ahler setting.