Bonatti's finiteness conjecture for chain-recurrence classes

Let a diffeomorphism be generic and far from homoclinic tangencies, and let chain-recurrence classes be the equivalence classes defined by mutual chain recurrence. Bonatti's conjecture. Such a diffeomorphism has finitely many chain-recurrence classes. This is a central finiteness problem for dynamics far from homoclinic tangencies; the source describes it as probably the main remaining open question in that setting, and the supplied excerpt gives no resolution.