Katok's exponential periodic-orbit growth conjecture for surface diffeomorphisms

Let $M$ be a compact surface, let $f\colon M\to M$ be a diffeomorphism, let $P_n(f)=\{x\in M:x=f^n(x)\}$, and let $h_{\operatorname{top}}(f)$ denote the topological entropy of $f$. Katok's conjecture. Is it true that

$$\limsup_{n\to +\infty}\frac{\sharp P_n(f)}{e^{n h_{\operatorname{top}}(f)}}>0?$$

for any (i) $C^r$ ($r>1$) or (ii) $C^\infty$ diffeomorphism $f$ of a compact surface? The conjecture asks for an exponential lower-growth estimate for periodic orbits at the topological entropy scale; the source does not state whether it has been resolved.