The form-type Calabi–Yau equation solvability conjecture

Let $M$ be a compact complex manifold with a balanced metric $\omega_0$ and a Hermitian metric $\alpha$, and let $h$ be a smooth function. Seek a function $\varphi$ and a constant $b$ such that the $(n-1)$st power of a Hermitian metric $\hat{\omega}$ is positive and satisfies

$$\hat{\omega}^n=e^{h+b}\alpha^n,$$ $$\hat{\omega}^{n-1}=\omega_0^{n-1}+\sqrt{-1}\,\partial\bar\partial(\varphi\alpha^{n-2})>0.$$

Form-type Calabi–Yau equation solvability conjecture. There exist a constant $b$ and a function $\varphi$ satisfying these equations. The equation is introduced as the form-type Calabi–Yau equation and is related to the problem of constructing balanced Chern–Ricci-flat metrics; the supplied text gives no resolution of this conjecture.