Ricci form of a Kaehler manifold

Question

Let $(M,\omega)$ be a Kaehler manifold with Kaehler form $\omega$. Does it exist another Kaehler form $\omega'$ in the same cohomology class of $\omega$ such that the Ricci form $Ric'$ of $\omega'$ is harmonic either for $\omega$ or for $\omega'$?

$$\omega' \in [\omega]$$

1) harmonic for $\omega$:   

$$\Delta_{\omega} Ric'=0$$

2) harmonic for $\omega'$:

$$\Delta_{\omega'} Ric'=0$$