Let $M,N$ be closed smooth manifolds. Let $f_0,f_1\colon M\to N$ be two smooth fibrations which are homotopic to each other in the class of smooth (equivalently, continuous) maps. Let $E^i_0, E^i_1$ be two flat vector bundles whose fibers over a point $x\in N$ are equal to the cohomology of the fibers $H^i(f_0^{-1}(x),\mathbb{R})), H^i(f_1^{-1}(x), \mathbb{R})$ respectively.
Is it true that $E^i_0, E^i_1$ are isomorphic as flat vector bundles?
