Converse torus conjecture for dynamical coherence

Let $M^3$ be a 3-dimensional manifold and let $f:M^3\rightarrow M^3$ be a partially hyperbolic diffeomorphism. A $cu$-torus is a 2-torus tangent to the center-unstable bundle $E^{cu}$, and an $sc$-torus is a 2-torus tangent to the stable-center bundle $E^{sc}$. Converse torus conjecture. If $f$ is not dynamically coherent, then it admits either a $cu$-torus or an $sc$-torus. The preceding theorem shows that a dynamically coherent partially hyperbolic diffeomorphism cannot have a compact leaf in its center-unstable foliation; this conjecture proposes the converse obstruction, relating failure of dynamical coherence to the existence of a torus tangent to one of the two mixed bundles.