Conjecture on topological triviality from irreducible partially hyperbolic dynamics

Let $X$ be a simply connected closed manifold or a simply connected finite simplicial complex, and let $p\colon E\to X$ be a fiber bundle with closed-manifold fiber $M$. Assume that $E$ admits a fiberwise irreducible partially hyperbolic diffeomorphism whose center distribution has dimension $1$ or $2$. Here irreducible means that the diffeomorphism satisfies the three conditions specified in the source: it does not fiber over a lower-dimensional topologically partially hyperbolic or Anosov diffeomorphism, this property persists under homotopy, and it persists under finite covers. Irreducible-center conjecture. The bundle $p\colon E\to X$ is topologically trivial. This is proposed as a simply connected-base generalization after the product/Hopf-fibration example shows that unrestricted fiberwise partial hyperbolicity is insufficient; the supplied text gives no resolution status.