Avila–Bochi conjecture on generic non-uniform hyperbolicity

Let $f$ be a $C^{1}$-generic diffeomorphism of a closed connected manifold. Let the Pesin region $\operatorname{Nuh}(f)$ be the set of points with no zero Lyapunov exponents, and let the Oseledets splitting be the Lyapunov splitting defined almost everywhere by Oseledets' theorem. A splitting is dominated when it satisfies the usual uniform domination inequality over the invariant set on which it is defined.

Avila–Bochi conjecture. Either all Lyapunov exponents of $f$ vanish almost everywhere, or the Pesin region $\operatorname{Nuh}(f)$ has full measure, $f$ is ergodic, and the Oseledets splitting extends to a global dominated splitting.

This conjecture generalizes generic results of Avila–Bochi and Mañé–Bochi on the alternatives between vanishing Lyapunov exponents and global domination. Its status is not resolved in the supplied source.