Pesin's nonzero center Lyapunov exponents conjecture

Let $M$ be a compact smooth Riemannian manifold, and let $\operatorname{Diff}^r_m(M)$ denote the $C^r$ diffeomorphisms preserving volume $m$. For $f\in\operatorname{Diff}^r_m(M)$ with $r>1$, Pesin's conjecture. Arbitrarily close to $f$ in $\operatorname{Diff}^r_m(M)$ there exists a diffeomorphism $g$ with nonzero Lyapunov exponents on a set of positive volume. Moreover, there is an open set $U$ containing $g$ and a residual subset of $U$ such that every $h$ in this subset has nonzero Lyapunov exponents on a set of positive measure. This predicts the prevalence of nonuniform hyperbolicity near volume-preserving partially hyperbolic systems; the supplied text does not indicate whether the conjecture has been resolved.