Let $E\to X$ be a holomorphic vector bundle over a Kähler manifold. The vanishing of the Atiyah class, $At(E)=0$, is equivalent to the existence of a holomorphic connection on $E$.

Moreover, it is well known that $At(E)=0$ implies that all the rational Chern classes of $E$, of degree at least one, vanish. Therefore, there is no topological obstruction to the existence of a flat connection on $E$.

There is a paper by I. Biswas analyzing the related question of the existence of a flat connection compatible with the holomorphic structure of $E$.

However I have been unable to find an answer to the more general question regarding the existence of a flat connection on $E$, not necessarily compatible with the holomorphic structure of $E$.

Notice however, that in the end this is equivalent to asking wether there exists a flat connection on $E$ compatible with an eventual change in the holomorphic structure of $E$.

So please, can anybody provide a counterexample or a reference proving that all holomorphic vector bundles $E\to X$ with $At(E)=0$ are flat.