The goal of this short article is to describe the local structure of the Teichmüller stack of [8] in the neighborhood of a Kähler point. In particular we show that at a generic Kähler point X, Catanese Kur=Teich question, when interpretated at the level of stacks, has an affirmative answer. The situation may be much more complicated if X is non-Kähler suggesting that Teichmüller spaces/stacks of non-Kähler manifold has a much richer geometry.