Pesin's density conjecture for non-uniformly hyperbolic conservative diffeomorphisms

Let $M$ be a compact smooth Riemannian manifold with $\dim M\geq 2$, let $r>1$, and let $f$ be a $C^r$ conservative diffeomorphism of $M$. Write $\operatorname{Diff}^r_m(M)$ for the space of $C^r$ volume-preserving diffeomorphisms, where $m$ is volume. A diffeomorphism has no zero Lyapunov exponents on a set when all its Lyapunov exponents are nonzero at every point of that set.

Pesin's density conjecture. Arbitrarily close to $f$ in $\operatorname{Diff}^r_m(M)$, there is a diffeomorphism $g\in \operatorname{Diff}^r_m(M)$ without zero Lyapunov exponents on a set of positive volume.

This asks whether non-uniformly hyperbolic behavior is dense among conservative diffeomorphisms. The paper presents it as a question formulated by Pesin; the supplied status evidence does not establish a resolution.