Bonatti's finiteness conjecture for tame diffeomorphisms

Let $M$ be a compact manifold, let $\operatorname{Tang}$ denote the set of diffeomorphisms exhibiting a homoclinic tangency, and let $\operatorname{Diff}^1(M)\setminus\overline{\operatorname{Tang}}$ be its complement away from the closure of tangencies. Bonatti's finiteness conjecture. In $\operatorname{Diff}^1(M)\setminus\overline{\operatorname{Tang}}$, there exists an open and dense subset of tame diffeomorphisms. Here, tameness refers to the finiteness property for chain-recurrence classes used in the surrounding discussion. The conjecture is motivated by finiteness results for quasi-attractors and is unresolved in the source.