Hurley's conjecture on quasi-attractors of generic points

Let $M$ be a compact manifold, let ${\rm Diff}^r(M)$ be the space of $C^r$ diffeomorphisms of $M$, and call a point generic when it belongs to the full-measure generic set under consideration. For $f\in {\rm Diff}^r(M)$ and $x\in M$, write $\omega(x)$ for the $\omega$-limit set of $x$. Hurley's conjecture. For $C^r$-generic diffeomorphisms, the $\omega$-limit sets of generic points in $M$ are quasi attractors. The source states that this conjecture was proved for the $C^1$ topology and remains open in more regular topologies.