A compact complex manifold is called a $\partial\bar{\partial}$-manifold if for every pair $p,q\in \mathbb{N}$, every smooth $d$-closed $(p,q)$-form $\eta$ on $X$, $\eta$ is $d$-exact iff $\partial$-exact iff $\bar{\partial}$-exact iff $\partial\bar{\partial}$-exact.
Classical Hodge theory shows all compact Kähler manifolds are $\partial\bar{\partial}$-manifolds, but not vice versa.
According to Theorem 5.22 of Real homotopy theory of Kähler manifolds (by Deligne et el.), given a bimeromorphic morphism $X\to Y$ of compact complex manifolds, if $X$ is a $\partial\bar{\partial}$-manifold, then we know $Y$ is also a $\partial\bar{\partial}$-manifold.
I wonder what happens in the other direction. Explicitly, if $M\to N$ is a finite covering of compact complex manifolds, and $N$ is a $\partial\bar{\partial}$-manifold, do we have that $M$ is a $\partial\bar{\partial}$-manifold?
This question is natural and valuable.
EDIT: I don't know the case of finite covering. But if we still consider a bimeromorphic morphism $f:X\rightarrow Y$. As far as I know, the converse of [DGMS] has been confirmed in the case $\text{dim}\leq 3$ (which is maybe the best possible result up to now) by S. Rao--S. Yang--X. Yang, Dolbeault cohomologies of blowing up complex manifolds. J. Math. Pures Appl. (9) 130 (2019), 68–92.
To prove this claim, they established Dolbeault and De Rham blow up formulae, a characteristic of the $\partial\bar{\partial}$-lemma also provides the energy. For more details, one can refer to Remark 4.6 in their article.