Volume-hyperbolicity dichotomy conjecture for diffeomorphisms

Let $M$ be a compact manifold, and let $\operatorname{Diff}^1(M)$ be the space of $C^1$ diffeomorphisms. A chain-recurrence class is a sink when it is represented by an attracting periodic orbit, and a class is volume hyperbolic when it has the stated dominated splitting with volume expansion on the $F$ bundle. Volume-hyperbolicity dichotomy conjecture. There exist two disjoint open sets $\mathcal{U}_1,\mathcal{U}_2$ whose union is dense in $\operatorname{Diff}^1(M)$ and which satisfy: Baire-generic diffeomorphisms in $\mathcal{U}_1$ have infinitely many sinks; and every $f\in\mathcal{U}_2$ is volume hyperbolic, meaning that each chain-recurrence class $\Lambda$ that is not a sink has a dominated splitting $T_\Lambda=E\oplus F$, with $F$ nontrivial and $|\det(Df^N_{|F})|>1$ on $\Lambda$ for some $N\geq1$. This is proposed as a characterization of the absence of the Newhouse phenomenon; no resolution is given in the source.