Gauduchon's conjecture on prescribed Chern-Ricci curvature

Let $M$ be a compact complex manifold, and let $\psi$ be a closed real $(1,1)$-form on $M$ satisfying

$$[\psi]=c_{1}^{BC}(M)\in H_{BC}^{1,1}(M,\mathbb{R}).$$

A Gauduchon metric is a metric $\omega$ satisfying

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

Gauduchon's conjecture. There exists a Gauduchon metric $\omega$ on $M$ such that

$$\operatorname{Ric}(\omega)=\psi,$$

where $\operatorname{Ric}(\omega)$ is the Chern-Ricci curvature, locally given by

$$\operatorname{Ric}(\omega)=-\sqrt{-1}\,\partial\overline{\partial}\log\omega^n.$$

The conjecture asks for prescribed Chern-Ricci curvature within the Bott–Chern first Chern class; the supplied text gives no evidence of a resolution.