Higher-smoothness entropy conjecture for cycles of basic sets

Let $f$ be a $C^r$-diffeomorphism of a manifold $M$, and let $p$ be a hyperbolic periodic point belonging to a cycle of basic sets having no dominated splitting of any index. Write $\Delta(f,p)$ for the entropy quantity associated with $f$ and $p$. Higher-smoothness entropy conjecture. There exists a diffeomorphism $g$, arbitrarily close to $f$ in the $C^r$-topology, with a horseshoe $K$ such that

$$h_{\operatorname{top}}(g,K) \geq \frac{\Delta(f,p)}{r}.$$

This conjecture proposes a higher-regularity analogue of the perturbative entropy result proved in the $C^1$ setting. The surrounding discussion attributes the expected extension to a program of Gourmelon and places it in the setting of cycles of basic sets; no resolution is given here.