Let $f\colon E\to B$ be a fiber bundle of compact manifolds with fiber $F$. Assume that the push-forward $Rf_*(\underline{\mathbb{F}})$ in the derived category of the constant sheaf with coefficients in a field $\mathbb{F}$ is isomoprhic (in the derived category) to the direct sum of its cohomology sheaves (with appropriate shifts). As far as I understand, this implies that the spectral sequence of this fibration degenerates at the second term.
My question is whether the converse is true: namely if the spectral sequence degenerates at the second term, does it imply that the push-forward of the constant sheaf is isomoprhic in the derived category to the direct sum of its cohomology sheaves?
