This is partly inspired by answers to the question:
https://mathoverflow.net/questions/42709/question-about-hodge-number/42735#42735 .
Is there a family of compact complex manifolds, where the general fibres are
Kähler, but for which $E_1$ degeneration of the Hodge to de Rham spectral sequence fails
at the special fibre? Or, even better, such that the special fibre has nonclosed
holomorphic forms? 

I feel like I should know the answer, but somehow I don't. All
the examples I know where the spectral sequence doesn't degenerate are nilmanifolds*,
so they aren't even homotopic to  Kähler manifolds by standard rational homotopy theoretic obstructions (e.g. they aren't formal).
Also the famous Hironaka example [Ann. Math 1962] won't work either, because
the special fibre is an algebraic variety, so the spectral sequence will degenerate
(by an argument that can found in Deligne [Théorème de Lefschetz...]).
Obviously, I haven't thought about this deeply enough, but perhaps someone else has**.

**Footnotes** 

*I was bit sloppy yesterday, since the examples I have in mind include
solvmanifolds. However, there are still topological obstructions to these being Kähler
due to Nori and myself.

** From the answers, I gather that the work of Popovici suggests that
there may be no counterexample.