Conjecture on topological triviality from low-dimensional center dynamics

Let $X$ be a closed manifold or finite simplicial complex, and let $p\colon E\to X$ be a fiber homotopically trivial bundle with closed-manifold fiber $M$. Assume that $E$ admits a fiberwise partially hyperbolic diffeomorphism whose center distribution has dimension $1$ or $2$. Low-dimensional-center conjecture. The bundle $p\colon E\to X$ is topologically trivial. This would extend the paper’s topological-triviality results for bundles admitting fiberwise hyperbolic dynamics to a broader partially hyperbolic setting; the supplied text gives no resolution status.