The balanced Calabi–Yau metric conjecture

Let $M$ be a compact complex manifold of complex dimension $n$ with vanishing first Bott–Chern class $c_1^{\mathrm{BC}}(M)=0$, and let $\omega$ be a balanced Hermitian metric, meaning that $d(\omega^{n-1})=0$. A balanced metric $\tilde{\omega}$ is required to satisfy

$$[\tilde{\omega}^{n-1}]=[\omega^{n-1}]\quad\text{in }H^{2n-2}(M,\mathbb{R}).$$

The balanced Calabi–Yau metric conjecture. There is a balanced metric $\tilde{\omega}$ with

$$[\tilde{\omega}^{n-1}]=[\omega^{n-1}]\quad\text{in }H^{2n-2}(M,\mathbb{R}),\qquad \operatorname{Ric}(\tilde{\omega})=0.$$

This conjecture seeks a canonical analogue of Ricci-flat Kähler metrics on non-Kähler Calabi–Yau manifolds. It would produce many balanced metrics whose Bismut connections have vanishing Ricci curvature, but its general solvability remains open.