  Given a factor code $\pi$ from a shift of finite type $X$ onto a sofic shift
$Y$, an ergodic measure $\nu$ on $Y$, and a function $V$ on $X$ with summable
variation, we prove an invariant upper bound on the number of ergodic measures
on $X$ which project to $\nu$ and maximize $h(\mu) + \int V d\mu$ among all
measures in the fiber $\pi^{-1}(\nu)$. If $\nu$ is fully supported, this bound
is the class degree of $\pi$. This generalizes a previous result for the
special case of $V=0$.
