Fujiki class $\mathcal C$ with a symplectic structure

Question

Recall that a compact complex manifold $X$ is said to be in Fujiki class $\mathcal C$ if there is a proper modification $\mu:\tilde X\to X$ such that $\tilde X$ is a compact Kähler manifold. If $X$ admits a symplectic structure, i.e. $X$ carries a closed nondegenerate 2-form $\omega$, then is $X$ Kähler?

Answer 1

If $X'$ is a Mukai flop of a compact hyper-Kähler manifold $X$, then $X'$ is in Fujiki class $\mathcal{C}$ and carries a holomorphic symplectic form $\sigma$. Taking the real part or the imaginary part of $\sigma$ gives a symplectic form on $X'$.

There exist however Mukai flops which are not Kähler, see e.g. this paper of Yoshioka, Section 4.4.