The balanced metric conjecture for compact complex manifolds satisfying the $\partial\overline{\partial}$-lemma

A compact complex manifold is a complex manifold that is compact, and a balanced metric on a complex manifold of complex dimension $n$ is a Hermitian metric whose fundamental $(1,1)$-form $\omega$ satisfies

$$d(\omega^{n-1})=0.$$

The $\partial\overline{\partial}$-lemma is the property that every $d$-closed form which is both $\partial$-exact and $\overline{\partial}$-exact is also $\partial\overline{\partial}$-exact.

Balanced metric conjecture. Each compact complex manifold $X$ satisfying the $\partial\overline{\partial}$-lemma admits a balanced metric.

This conjecture asks whether the $\partial\overline{\partial}$-lemma, a strong condition controlling complex and cohomological properties, guarantees the existence of a balanced metric. The supplied text does not state whether the conjecture is resolved.