Finiteness of high-entropy topological homoclinic classes

Let $M$ be a closed surface and let $f:M\rightarrow M$ be a $\mathcal{C}^{\infty}$ local diffeomorphism. For an $f$-invariant measure $\mu$, let $\operatorname{HC}(\mu)$ denote the topological homoclinic class associated with the measured homoclinic class of $\mu$, and let $h_{top}(f,K)$ denote the topological entropy of $f$ on an $f$-invariant set $K$. Finiteness conjecture. For every $\chi>0$, there are only finitely many topological homoclinic classes $\operatorname{HC}(\mu)$ satisfying

$$h_{top}(f,\operatorname{HC}(\mu))>\log\deg(f)+\chi.$$

This conjecture generalizes to arbitrary topological degree the spectral decomposition proved by Buzzi, Crovisier, and Sarig. The supplied status evidence indicates that it has been resolved.