Palis's weak conjecture on generic Morse-Smale or homoclinic dynamics

Let $M$ be a compact manifold and let $\operatorname{Diff}^r(M)$ be the space of $C^r$-diffeomorphisms, where $r\geq 1$, endowed with the $C^r$-topology. Let $\mathcal{MS}$ denote the set of Morse-Smale diffeomorphisms and let $\mathcal{I}$ denote the set of diffeomorphisms having a transverse homoclinic intersection. Palis's weak conjecture. The space $\operatorname{Diff}^r(M)$ contains a dense open set which decomposes as the union $\mathcal{MS}\cup\mathcal{I}$ of two disjoint open sets. One interest of the conjecture is that it gives information on a large (open and dense) set of diffeomorphisms.