$M$ is non-Kähler complex manifold. Assume that $\omega$ is $\partial$-exact and $\bar\partial$-exact $(p,q)$-form.

> **Question**. Is $\omega$ also $\partial\bar\partial$-exact?

Based on fabulous David Speyer's answer to [this MO question][1] I suspect that the answer is elementary or very hard.


  [1]: http://mathoverflow.net/questions/59479/partial-bar-partial-lemma-for-contractible-domains