Generic stable ergodicity for tuples of volume-preserving diffeomorphisms

Let $M$ be a closed manifold with a volume, let $k\ge 1$, and let $\operatorname{Diff}^k_{\operatorname{vol}}(M)$ denote the space of $C^k$ volume-preserving diffeomorphisms of $M$. A tuple $(f_1,\ldots,f_m)$ is stably ergodic when its associated random dynamical system remains ergodic under sufficiently small perturbations. Generic stable ergodicity conjecture. For each closed manifold $M$ with volume and regularity $k\ge 1$, there exists $m$ such that the space of stably ergodic tuples $(f_1,\ldots,f_m)$ is open and dense in

$$\left(\operatorname{Diff}^k_{\operatorname{vol}}(M)\right)^m.$$

The conjecture asserts that a small, possibly minimal, amount of randomness should generically ensure robust ergodic and statistical properties; its resolution would make exponential mixing a generic property for random dynamical systems.