Avila–Crovisier–Wilkinson conjecture on stable ergodicity and dominated splittings

Let $M$ be a compact manifold, let $\omega$ be a volume form, and let $\operatorname{Diff}^r_\omega(M)$ denote the space of $C^r$ volume-preserving diffeomorphisms, with $r>1$. A diffeomorphism is stably ergodic if its volume is ergodic and this property persists under $C^1$ perturbations, and a dominated splitting is a nontrivial invariant dominated splitting of the tangent bundle. Avila–Crovisier–Wilkinson conjecture. For $r>1$, the sets of stably ergodic diffeomorphisms and of those having a dominated splitting have the same $C^1$-closure in $\operatorname{Diff}^r_\omega(M)$. This conjecture relates stable ergodicity to dominated splittings in conservative dynamics; the source presents it as open and gives no resolution.