The Ricci flat LCK metrics

Question

Let $M$ be a manifold with zero first Chern class of cohomology $c_1(M)=0$, and $\omega$ be a LCK metric, 

$$d_{\theta}\omega =d\omega +\theta \wedge \omega =0$$

$$d\theta=0$$

Does it exist a metric $\omega'$ in the same class of Novikov cohomology,

$$\omega'=\omega +d_{\theta}\alpha$$

such that the Ricci curvature of $\omega'$ is zero?

$$\rho(\omega')=0$$

For $\theta=0$, it is the theorem of Yau for the Calabi-Yau manifolds.