I vaguely remember that I once attended a seminar or conference talk in which it was mentioned that the following question is open.

*Is there a (smooth) surface bundle over a surface $\Sigma_h \to E \to \Sigma_g$ that does not admit a flat structure? (Equivalently, one can ask whether there is a homomorphism $\pi_1(\Sigma_g) \to \pi_0 \text{Diff}(\Sigma_h)$ that cannot be lifted to the group of diffeomorphisms.)*

What is the status of this question, and could somebody point out a reference that mentions this as an open question? Finally, is something known for small values of $g,h$ (but still $g,h \geq 2$)?

(In case you do not understand what I am asking, I can very much recommend reading the last chapter of Morita's book *Geometry of Characteristic Classes*.)