Upper semicontinuity of high-entropy SRB measures

Let $f$ be a $C^\infty$ surface diffeomorphism and let $b>0$. Let $N^{\mathrm{SRB}}_g(b)$ denote the number of SRB measures of $g$ with metric entropy larger than $b$. SRB-measure conjecture. There is a $C^\infty$ neighborhood $\mathcal U$ of $f$ such that, for every $g\in\mathcal U$,

$$0\le N^{\mathrm{SRB}}_g(b)\le N^{\mathrm{SRB}}_f(b)<\infty.$$

The conjecture proposes local upper semicontinuity and finiteness of SRB measures above any positive entropy threshold for smooth surface diffeomorphisms. It extends the known finiteness result for SRB measures whose positive Lyapunov exponents exceed a prescribed threshold.