Measure classification conjecture for stationary measures of symplectic diffeomorphism groups

Let $Q$ be a compact symplectic manifold, and let $\mu$ be a probability measure on the group of symplectic diffeomorphisms of $Q$ that satisfies uniform expansion on all isotropic subspaces of $TQ$. Let $\nu$ be a $\mu$-stationary measure on $Q$ with no zero Lyapunov exponents.

Measure classification conjecture. The measure $\nu$ is a $\mu$-invariant measure on a smooth submanifold $M$ of $Q$, where the case $M=Q$ is allowed.

This proposes a geometric classification of stationary measures under the stated expansion and nonzero-exponent hypotheses. The surrounding results establish measure-rigidity and classification theorems under related assumptions, but the supplied text does not state whether this formulation has been proved or remains open.