Let $(M, \omega_M, J_M)$ and $(N, \omega_N, J_N)$ be compact Kähler manifolds. Denote $g_M=\omega_M(\cdot, J_M\cdot)$ and $g_N=\omega_N(\cdot, J_N\cdot)$.

Assume there is a diffeomorphism $\phi:M\to N$ such that $\phi^*(g_N)=g_M$. Is there a diffeomorphism $\psi:M\to N$ such that $\psi^*(\omega_N)=\omega_M$? What if we additionally assume that $\phi^*[\omega_N]=[\omega_M]\in H^2(M, \mathbb{R})$?