The robust polynomial mixing conjecture for dominated splittings

Let $r\ge1$. Consider diffeomorphisms with a dominated splitting, together with non-uniform expansion and non-uniform contraction, and with neither a uniformly expanding nor a uniformly contracting subbundle. Robust polynomial mixing conjecture. There are examples of $C^r$ open subsets of diffeomorphisms having mixing physical measures which do not mix exponentially. This would establish robust examples with slower-than-exponential mixing in the absence of uniformly expanding or contracting subbundles, complementing the paper's non-robust examples with polynomial mixing rates.