Folklore conjecture on maximum measures for surface diffeomorphisms

Let $f:M\to M$ be a $C^\infty$ surface diffeomorphism, and let $h_{\operatorname{top}}(f)$ denote its topological entropy. A maximum measure is an invariant probability measure whose measure-theoretic entropy equals $h_{\operatorname{top}}(f)$. Folklore conjecture. If

$$h_{\operatorname{top}}(f)>0,$$

then $f$ has finitely many maximum measures. This is the surface-diffeomorphism analogue of the corresponding result for smooth interval maps; it remains open, although related results are known for toy models.