Generalized local entropy estimate for $^r$ maps

Let $r>1$, let $M$ be a compact manifold of dimension $d$, and let $T:M\to M$ be a $^r$ map. For nonintegral $r$, assume $T\in \mathcal{C}^{\lfloor r\rfloor}$ and $D^{\lfloor r\rfloor}T$ is $(r-\lfloor r\rfloor)$-Hölder. Let $\mu$ be an invariant measure and fix $\gamma>0$. For an ergodic measure $\nu$, write $\Sigma^+\chi(\nu)$ for the sum of its positive Lyapunov exponents, and let $\overline{\Sigma^+\chi}(\mu)$ denote the corresponding upper-semicontinuous quantity used in the source. Generalized local entropy conjecture. There exist $\delta_{\mu}>0$ and $\epsilon_{\mu}>0$ such that, for every ergodic measure $\nu$ satisfying $\operatorname{dist}(\nu,\mu)<\delta_{\mu}$,

$$h^{New}(M\mid\nu,\epsilon_{\mu})\leq \frac{\overline{\Sigma^+\chi}(\mu)-\Sigma^+\chi(\nu)}{r-1}+\gamma.$$

This would extend the preceding surface-diffeomorphism estimate to arbitrary dimensions, intermediate regularity, and noninvertible maps. The source presents it as a conjectural extension and gives no resolution status.