HHU ergodicity conjecture for conservative partially hyperbolic diffeomorphisms

Let $f: M^3\rightarrow M^3$ be a $C^r$ ($r > 1$) conservative partially hyperbolic diffeomorphism of a closed 3-manifold that fails to be ergodic. Here, $E^s$ and $E^u$ denote the stable and unstable bundles of the partially hyperbolic splitting, and a mapping torus is the manifold obtained from a self-diffeomorphism of $\mathbb{T}^2$ by identifying the ends of $\mathbb{T}^2\times[0,1]$ using that diffeomorphism. HHU ergodicity conjecture. There exists an embedded 2-torus tangent to $E^s \oplus E^u$. Moreover, under orientability assumptions, the manifold $M^3$ must be one of: the 3-torus $\mathbb{T}^3$; the mapping torus of $-id: \mathbb{T}^2 \rightarrow \mathbb{T}^2$; or the mapping torus of an Anosov diffeomorphism on $\mathbb{T}^2$. This conjecture seeks a complete characterization of the obstructions to ergodicity for conservative partially hyperbolic diffeomorphisms in dimension three, where prior work established that ergodicity is prevalent but not that all non-ergodic examples have the stated geometric form.