Bonatti's robust-cycle denseness conjecture

Let $M$ be a smooth manifold, and let $C^1$-diffeomorphisms of $M$ be considered with the $C^1$ topology. A diffeomorphism has a $C^1$-robust cycle if it exhibits either a homoclinic tangency or a heterodimensional cycle that persists under $C^1$-small perturbations. Bonatti's robust-cycle denseness conjecture. Every $C^1$-diffeomorphism can be $C^1$-approximated either by a hyperbolic diffeomorphism satisfying the Axiom A and no-cycle property, or by a diffeomorphism exhibiting a $C^1$-robust cycle. The results of the paper and its preceding work are described as supporting this conjecture, but the source does not state that it has been resolved.