  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.
