Tameness of one-dimensional partially hyperbolic diffeomorphisms in dimension three

Let $M$ be a compact $3$-manifold, and let ${\cal P}{\cal H}(M)$ be the $C^1$-open set of partially hyperbolic diffeomorphisms of $M$ admitting a dominated splitting

$$E^s\oplus E^c\oplus E^u,$$

where all three bundles have dimension $1$. Tameness conjecture. The open set ${\cal P}{\cal H}(M)$ does not contain any wild diffeomorphism; equivalently, every generic diffeomorphism in ${\cal P}{\cal H}(M)$ is tame. The source says that whether such dominated splittings prevent wild dynamics is unknown, even in dimension $3$.