Conjecture on stable ergodicity and dominated splittings

Let $M$ be the connected manifold in the source, let $r>1$, and consider the volume-preserving diffeomorphism space $\operatorname{Diff}^r_m(M)$ with its $C^1$ topology. A diffeomorphism is stably ergodic if it is ergodic and remains ergodic under all sufficiently small perturbations in the relevant topology; a non-trivial dominated splitting is a dominated invariant splitting of the tangent bundle that is not trivial. Stable ergodicity–dominated splitting conjecture. The sets of stably ergodic diffeomorphisms and of diffeomorphisms having a non-trivial dominated splitting have the same $C^1$-closure in $\operatorname{Diff}^r_m(M)$, for $r>1$. This conjecture proposes that, for smoother volume-preserving systems, dominated splitting characterizes stable ergodicity at the level of $C^1$-closures. The source presents it as a conjecture and gives no resolution status.