Rodriguez Hertz–Rodriguez Hertz–Ures conjecture on dynamical coherence

Let $M$ be a compact connected $3$-manifold without boundary, and let $f:M\to M$ be a partially hyperbolic diffeomorphism, meaning that there are $N>0$ and a $Df$-invariant continuous splitting

$$TM=E^s\oplus E^c\oplus E^u$$

into one-dimensional subbundles satisfying the stated partial hyperbolicity inequalities. Write $E^{cs}=E^s\oplus E^c$ and $E^{cu}=E^c\oplus E^u$. The diffeomorphism is dynamically coherent if there exist invariant foliations tangent to $E^{cs}$ and $E^{cu}$. Rodriguez Hertz–Rodriguez Hertz–Ures conjecture. If $f$ has no periodic two-dimensional torus tangent to $E^{cs}$ or $E^{cu}$, then $f$ is dynamically coherent. The conjecture identifies periodic two-dimensional tori as the unique obstruction to dynamical coherence for partially hyperbolic diffeomorphisms in dimension three. In the source, it is proved for manifolds with virtually solvable fundamental group; the general three-dimensional case remains open.