Hertz–Hertz–Ures weak ergodicity conjecture

Consider an orientable 3-manifold and a conservative partially hyperbolic diffeomorphism on it. Hertz–Hertz–Ures weak ergodicity conjecture. The only orientable 3-manifolds admitting a non-ergodic conservative partially hyperbolic diffeomorphism are the 3-torus, the mapping torus of $-\operatorname{id}:\mathbb T^2\to\mathbb T^2$, and mapping tori of hyperbolic automorphisms of the 2-torus. This is a weaker manifold-level consequence of the strong non-ergodicity conjecture; the source gives no resolution.