Abdenur–Bonatti–Crovisier transitivity conjecture for homoclinic classes

Let $M$ be a compact connected boundaryless manifold of dimension $d$, and let $\operatorname{Diff}^{1}(M)$ be the space of diffeomorphisms of $M$ with the $C^1$ topology. A homoclinic class $H(p,f)$ is the closure of the transverse intersections between the stable and unstable manifolds of the orbit of a hyperbolic periodic point $p$ of $f$. Abdenur–Bonatti–Crovisier conjecture. There exists a residual set $\mathcal{R}$ of $\operatorname{Diff}^{1}(M)$ such that if $f\in\mathcal{R}$ admits a homoclinic class with nonempty interior, then $f$ is transitive. The conjecture concerns whether a homoclinic class with nonempty interior forces generic global transitivity; it was posed in the cited work and also appeared as Problem 1 in the cited reference.