Stable ergodicity from positive metric entropy

Let $M$ be a smooth compact manifold, let $m$ be a smooth volume measure, and let $\operatorname{Diff}^{1}_{m}(M)$ denote the volume-preserving $C^{1}$ diffeomorphisms. Stable ergodicity conjecture from positive entropy. Generically in $\operatorname{Diff}^{1}_{m}(M)$, if $f$ has positive metric entropy with respect to Lebesgue measure, then $f$ is stably ergodic. This proposes dominated splitting as a generic mechanism for stable ergodicity, but the statement is presented as a conjecture without a resolution in the source.