Let $f:X\to Y$ be a surjective holomorphic map between two compact complex manifolds. Suppose that $R^if_*\mathbb{Q}_X$ are locally constant sheaves over $Y$ for all $i$, and $f$ is homotopic to a $C^{\infty}$-fiber bundle.
Q: Is it true that $f$ is a fibration, in particular, all the fibers of $f$ are homotopic to each other?
Any related reference is wellcome.