Symplectic orbit-closure and stationary-measure stratification conjecture

Let $Q$ be a compact symplectic manifold, and let $\mu$ be supported on a group $\Gamma_\mu$ of smooth symplectomorphisms of $Q$. Assume that $\mu$ has uniform expansion on every isotropic subspace of $T_qQ$, for every $q\in Q$, and uniform gaps in dimension $\tfrac12\dim Q$. Symplectic orbit-closure conjecture. Every $\mu$-stationary measure $\nu$ on $Q$ is supported on a manifold-stratified space $S_\mu\subseteq Q$, with symplectic open stratum $S_\mu^{\circ}$, and $\nu$ is the volume measure on $S_\mu^{\circ}$. Furthermore, for every $q\in Q$, the orbit closure $\overline{\Gamma_\mu\cdot q}$ is such a space $S_q\subseteq Q$, and the empirical measures of $q$ converge to the smooth volume on $S_q^{\circ}$. This is presented as a natural extension of an earlier finite-or-uniform measure-rigidity result and as a question appearing in the cited work; the stated general case remains open.