Burguet's periodic-point growth conjecture for surface diffeomorphisms

Let $f$ be a $C^r$ diffeomorphism, with $r>1$, on a compact surface $M$. Write $h_{\rm top}(f)$ for its topological entropy and $\lambda^{+}(f)$ for the relevant positive Lyapunov-exponent bound. For $0<\delta<h_{\rm top}(f)$, let $\mathcal{P}^{n}_{\delta}$ be the set of $n$-periodic points whose Lyapunov exponents are $\delta$-away from zero. Burguet's conjecture. If

$$h_{\rm top}(f)>\frac{\lambda^{+}(f)}{r},$$

then

$$h_{\rm top}(f)=\limsup_{n\to+\infty}\frac{1}{n}\log \#\mathcal{P}^{n}_{\delta}.$$

This extends Burguet's result in the $C^{\infty}$ setting to finite differentiability. The conjecture concerns whether the exponential growth rate of hyperbolic periodic points recovers the topological entropy under the stated entropy and Lyapunov-exponent condition.