The balanced Gauduchon conjecture for Ricci-flat metrics

Let $X$ be a compact complex manifold of dimension $n$ with holomorphic volume form $\Omega$. Let $\omega$ be a Hermitian metric satisfying

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

and denote its balanced class by $[\omega^{n-1}]\in H_{\rm BC}^{n-1,n-1}(X,\mathbb{R})$. The balanced Gauduchon conjecture. There exists a balanced metric $\tilde{\omega}$ with $\tilde{\omega}^{n-1}\in[\omega^{n-1}]$ such that ${\rm Ric}_{\tilde{\omega}}=0$.

This is a balanced analogue of Yau's theorem for non-Kähler Calabi–Yau manifolds. It is known when the background manifold $X$ admits a Kähler metric, and other constructions are known, but the conjecture remains open in general.