Hertz–Hertz–Ures Anosov-torus conjecture

Let $M$ be a 3-manifold and let $f:M\to M$ be a partially hyperbolic diffeomorphism with invariant bundles $E^u$, $E^c$, and $E^s$. An embedded torus is periodic if $f^k(S)=S$ for some positive integer $k$, and incompressible if its fundamental group injects into that of $M$. Hertz–Hertz–Ures's Anosov-torus conjecture. For every non-ergodic partially hyperbolic diffeomorphism on a 3-manifold $M$, there is an embedded, periodic, incompressible torus tangent to $E^u\oplus E^s$. Such a torus is an Anosov torus and would force $M$ to be one of the manifolds $M_B$ described above. The conjecture remains open.