Sharpness of the entropy threshold for infinitely many ergodic measures of maximal entropy

Let $f$ be a $C^r$ diffeomorphism of a closed surface, where $1<r<\infty$, and let $\lambda_{\min}(f)$ denote the minimum expansion rate appearing in the entropy estimates. A measure maximizing entropy is a measure $bc$ with $h(f,\mu)=h_{\operatorname{top}}(f)$; it is ergodic when it is ergodic for $f$. Sharpness conjecture. For any $\varepsilon>0$ and any $1<r<\infty$, there is a $C^r$ surface diffeomorphism $f$ with

$$h_{\operatorname{top}}(f)>\frac{\lambda_{\min}(f)}r-\varepsilon$$

and infinitely many ergodic measures maximizing entropy. This would show that the threshold in the finiteness theorem is sharp; the conjecture remains open.