Arnold-Kolmogorov typicality conjecture for finiteness of attractors

Let $M$ be a compact manifold. A property is Arnold–Kolmogorov typical if it holds for Lebesgue almost every small parameter in a Baire generic set of smooth families of dynamics. Finiteness-of-attractors conjecture. Typically, in the sense of Arnold–Kolmogorov, a diffeomorphism of $M$ has finitely many topological attractors, and hence finitely many sinks. This conjecture belongs to a broader program asserting typical finiteness of attractors; the source also mentions related formulations by Tedeschini-Lalli and Yorke, Palis and Takens, and Palis. Its resolution is not supplied in the source.