The balanced Chern–Ricci-flat metric conjecture

Let $M$ be a compact complex manifold with vanishing Bott–Chern first Chern class $c_1^{BC}=0$ and a balanced metric $\omega_0$. A Hermitian metric is balanced if $d(\omega^{n-1})=0$, and $Ric(\omega)$ denotes its Chern–Ricci form. Balanced Chern–Ricci-flat metric conjecture. There exists a balanced metric $\hat{\omega}$ such that

$$[\hat{\omega}^{n-1}]=[\omega_0^{n-1}]$$

in $H^{2n-2}(M,\mathbb{R})$ and $Ric(\hat{\omega})=0$. This conjecture asks for a Chern–Ricci-flat representative in every balanced cohomology class on a compact complex manifold with $c_1^{BC}=0$; the supplied text attributes it to Tosatti–Vezzoni and gives no resolution.