High-entropy finiteness of maximal-entropy measures for finite-regularity local diffeomorphisms

Fix $r>1$, and let $M$ be the underlying compact manifold and $f:M\rightarrow M$ a $\mathcal{C}^r$ local diffeomorphism. Write $\lVert f\rVert_{\mathcal{C}^1}$ for the $\mathcal{C}^1$ norm of $f$. Finite-regularity finiteness conjecture. There exists a constant

$$C=C(\lVert f\rVert_{\mathcal{C}^1},r)>0$$

depending only on the $\mathcal{C}^1$ norm of $f$ and $r$, such that if

$$h_{top}(f)>C,$$

then $f$ admits only finitely many ergodic measures of maximal entropy. This is posed as a possible extension of the paper's smooth setting to lower regularity; the supplied material does not establish it or provide evidence of resolution.