Existence of a measure of maximal entropy above the entropy threshold

Let $f$ be a $C^r$ surface diffeomorphism, where $1<r<\infty$, and let $h_{\operatorname{top}}(f)$ be its topological entropy. A measure maximizing entropy is an invariant probability measure $\mu$ satisfying $h(f,\mu)=h_{\operatorname{top}}(f)$. Existence conjecture. For any $1<r<\infty$, every $C^r$ surface diffeomorphism satisfying

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

has at least one measure maximizing entropy. Examples without such a measure are known below, but the asserted threshold remains open.