Relative uniqueness conjecture for maximal-pressure measures

Let $(X,Y,\pi)$ be a factor triple, let $\nu$ be a fully supported ergodic measure on $Y$, and let $c_\pi$ denote the class degree of the factor map. For a function $f\in \operatorname{Bow}(X)$, consider the ergodic invariant measures on $X$ that project to $\nu$ and have maximal pressure for $f$ in the fibre $\pi^{-1}\{\nu\}$. Relative maximal-pressure conjecture. The number of these measures is at most $c_\pi$. The absolute analogue follows from the uniqueness of equilibrium states for functions in $\operatorname{Bow}(X)$; this conjecture asks for the corresponding relative statement over a fully supported ergodic factor measure.