Berger's blender conjecture for entropy-dominated hyperbolic basic sets

Let $\mathcal{F}$ be a $C^r$-local diffeomorphism of a manifold $\mathcal{M}$, for $r \ge 2$, and let $\mathcal{K}$ be a hyperbolic basic set for $\mathcal{F}$. Write $\mathcal{E}^s$ for its stable bundle and define

$$m(D\mathcal{F}):= \min_{z \in \mathcal{K},\,u \in \mathcal{E}^s_z,\,\lVert u\rVert=1}\lVert D_z\mathcal{F}(u)\rVert.$$

Assume that the topological entropy $h_\mathcal{F}$ of $\mathcal{F}|\mathcal{K}$ satisfies

$$h_\mathcal{F}>\operatorname{dim}\mathcal{E}^s\,\lvert\log m(D\mathcal{F})\rvert.$$

Berger's conjecture. There exists a $C^r$-neighborhood $\mathcal{U}$ of $\mathcal{F}$ and an infinite-codimensional subset $\mathcal{N}\subset\mathcal{U}$ such that, for every $\widetilde{\mathcal{F}}\in\mathcal{U}\setminus\mathcal{N}$, the continuation $\widetilde{\mathcal{K}}$ of $\mathcal{K}$ is a $C^r$-blender.

This conjecture asks when entropy and contraction along the stable bundle force blender behavior; the paper presents it as a fundamental problem, and the supplied text gives no resolution of the full statement.