Bonatti's finite chain-recurrence classes conjecture

Let $M$ be the compact manifold underlying ${\operatorname{Diff}}^1(M)$, and let $\overline{\rm HT}$ denote the closure of the set of diffeomorphisms with homoclinic tangencies. Bonatti's conjecture. There exists a dense $G_\delta$ subset in

$${\operatorname{Diff}}^1(M)\setminus \overline{\rm HT}$$

of diffeomorphisms having only finitely many chain-recurrence classes. This predicts finiteness of the chain-recurrence decomposition generically away from homoclinic tangencies; the supplied text does not state whether it has been resolved.