Let $p\colon E\to B$ be a fibration with fibres simply connected and homotopy equivalent to a compact CW-complex. Must $p_*\colon H_3(E;\mathbb{Q})\to H_3(B;\mathbb{Q})$ be surjective?

COMMENTS. Yes if (EDIT) $B$ is simply connected, even if the fibre is not compact but just finite-dimensional. In general, finite-dimensionality is not enough: consider the homotopy fibre sequence $\mathbb{R}^3\setminus\mathbb{Z}^3\to T^3\setminus\mathrm{point}\to T^3$.

MOTIVATION. If $p_*\colon H_3(E;\mathbb{Q})\to H_3(B;\mathbb{Q})$ is surjective, then any bundle gerbe over $p\colon E\to B$ is rationally trivial, cf. M. Murray, D. Stevenson, A note on bundle gerbes and infinite-dimensionality (http://arxiv.org/abs/1007.4922).