On the cardinality of measures of maximal relative entropy for smooth skew products

Let $\mathrm{\Omega }$ and $M$ be compact smooth manifolds and let $\mathrm{\Theta }:\mathrm{\Omega }\times M\to \mathrm{\Omega }\times M$ be a ${\mathcal{C}}^{1+\alpha }$ skew-product diffeomorphism over a transitive Anosov base. We show that $\mathrm{\Theta }$ has at most countably many ergodic hyperbolic measures of maximal relative entropy. When $\dim M=2$, if $\mathrm{\Theta }$ has positive relative topological entropy, then $\mathrm{\Theta }$ has at most countably many ergodic measures of maximal relative entropy.