Weak Ergodicity Conjecture for partially hyperbolic diffeomorphisms on 3-manifolds

Let $M^3$ be an orientable closed 3-manifold and let $f:M^3\to M^3$ be a $C^r$ ($r>1$) non-ergodic partially hyperbolic diffeomorphism. Weak Ergodicity Conjecture. Then $M^3$ must be one of the following: the 3-torus $\mathbb T^3$; the mapping torus of $-\operatorname{id}:\mathbb T^2\to\mathbb T^2$; or a mapping torus of an Anosov diffeomorphism of $\mathbb T^2$. This conjecture seeks a topological characterization of 3-manifolds admitting non-ergodic partially hyperbolic dynamics; its general status is not resolved in the supplied text.