Tosatti–Weinkove conjecture for the modified Hermitian Monge–Ampère equation

Let $M$ be a compact complex manifold with a Hermitian metric $\omega_0$ and a Gauduchon metric $\omega$, and let $F$ be a smooth function. For a smooth function $u$, define the $(n-1,n-1)$-form

$$\Phi_u=\omega_0^{n-1}+\sqrt{-1}\,\partial\overline{\partial}u\wedge\omega^{n-2}+\operatorname{Re}\left(\sqrt{-1}\,\partial u\wedge\overline{\partial}(\omega^{n-2})\right).$$

Tosatti–Weinkove's conjecture. There exists a unique pair $(u,b)$, with $u$ a smooth function on $M$ and $b$ a constant, such that

$$\det(\Phi_u)=e^{F+b}\det(\omega^{n-1}),$$

with $\Phi_u>0$ and $\sup_M u=0$. The equation is a modification of the Monge–Ampère equation for Gauduchon metrics and would imply Gauduchon's full conjecture; the source notes that Popovici posed the same assertion as a question.