Robust transitivity versus trapping regions for partially hyperbolic diffeomorphisms

We show that among partially hyperbolic diffeomorphisms with one dimensional center there is a $C^1$-open and dense subset for which either there is a proper quasi-attractor or both strong foliations are minimal. The same result holds among volume-preserving partially hyperbolic diffeomorphisms and has consequences about robust transitivity beyond the conservative setting. The proofs involve a careful study of minimal ${\mathcal{W}}^u$-saturated sets and interact with recent results in the subject.