Hertz–Hertz–Ures ergodic conjecture for three-dimensional partially hyperbolic systems

Let $f$ be a conservative partially hyperbolic diffeomorphism of a 3-manifold. Hertz–Hertz–Ures ergodic conjecture. If $f$ is non-ergodic, then there is a 2-torus tangential to $E^s\oplus E^u$. In particular, the only orientable 3-manifolds that admit a non-ergodic conservative partially hyperbolic diffeomorphism are the 3-torus $\mathbb{T}^3$, the mapping torus of ${\rm -Id}$, or the mapping torus of a hyperbolic automorphism of the 2-torus. This conjecture concerns the proposed classification of non-ergodic conservative partially hyperbolic diffeomorphisms in dimension three; the supplied text gives no resolution status.