Palis's finiteness and prevalence conjecture for ergodic attractors

Let $M$ be a compact manifold and let ${\rm Diff}^r(M)$ denote the space of $C^r$ diffeomorphisms of $M$. An ergodic attractor is an attractor supporting an ergodic invariant probability measure whose support is the attractor and whose basin has positive Lebesgue measure. Palis's conjecture. There is a dense set ${\cal D}\subset {\rm Diff}^r(M)$ such that for any $f\in\cal D$, $f$ has only finitely many ergodic attractors, and the union of the basins of attractors forms a full Lebesgue measure set in $M$. The conjecture has been proved for the $C^1$ topology and remains open in more regular topologies, according to the source.