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We consider partially hyperbolic diffeomorphisms $f$ with a one-dimensionalcentral direction such that the unstable entropy exceeds the stable entropy.Our main result proves that such maps have a finite number of ergodic measuresof maximal entropy. Moreover, any $C^{1+}$ diffeomorphism near $f$ in the $C^1$topology possesses at most the same number of ergodic measures of maximalentropy. Our contribution is in extending the findings in \cite{BCS22} toarbitrary dimensions. We believe our technique, essentially distinct from theone in that paper, is robust and may find applications in further contexts.

Finite Measures of Maximal Entropy for an Open Set of Partially Hyperbolic Diffeomorphisms