The generic dichotomy between zero exponents and nonuniform Anosov ergodicity

Let $M$ be the compact manifold equipped with volume measure $m$, and let $\mathrm{Diff}_m^1(M)$ denote the space of $C^1$ volume-preserving diffeomorphisms. A diffeomorphism is nonuniformly Anosov when it is nonuniformly hyperbolic and has a global dominated splitting separating the positive Lyapunov exponents from the negative ones. Generic nonuniform Anosov dichotomy. For generic $f\in \mathrm{Diff}_m^1(M)$, either $f$ has all exponents zero at Lebesgue almost every point, or $f$ is ergodic and nonuniformly Anosov. The conjecture aims to separate the nonuniformly hyperbolic setting from the world of maps with zero exponents; the paper presents this as an optimistic formulation of the question, and the general dichotomy remains open.