Continuity of topological entropy for partially hyperbolic diffeomorphisms with one-dimensional center

Let $M$ be a compact manifold, and let $f$ be a $C^1$ partially hyperbolic diffeomorphism of $M$ with a decomposition $TM=E^s\oplus E^c\oplus E^u$ into invariant continuous sub-bundles, where $E^c$ has dimension one. Continuity conjecture. The topological entropy is continuous on the space of $C^1$ partially hyperbolic diffeomorphisms with the dimension of the center equal to one. Uniformly hyperbolic diffeomorphisms have locally constant topological entropy, whereas continuity is known to fail in some examples with two-dimensional center; the one-dimensional-center case remains the focus of this conjecture.