Generic symplectic Oseledets splitting and ergodicity conjecture

Let $M$ be the symplectic manifold from the surrounding discussion, let $\omega$ be its symplectic form, and let $\operatorname{Diff}^1_\omega(M)$ denote the space of $C^1$ symplectomorphisms. For a regular point $x$, write the Oseledets splitting as $T_xM=E^+(x)\oplus E^0(x)\oplus E^-(x)$, where the bundles correspond respectively to positive, zero, and negative Lyapunov exponents. A splitting is dominated if it is a globally dominated invariant splitting.

Generic symplectic Oseledets splitting conjecture. For a generic $f\in \operatorname{Diff}^1_\omega(M)$ either all Lyapunov exponents vanish almost everywhere, or else the Oseledets splitting extends to a globally dominated (partially hyperbolic) splitting and in this case $f$ is ergodic.

This conjecture proposes a dichotomy between an elliptic regime and a globally dominated, ergodic regime for generic symplectomorphisms. The preceding discussion records a related theorem of Bochi with a weaker alternative, while the asserted global extension together with ergodicity is not established here.