Let $X$ be a compact complex manifold in Fujiki class $\mathcal C$, that is bimeromorphic to a compact Kähler manifold, let $T$ be a Kähler current of $X$, then we have the De Rham class $[T]\in H^{1,1}(X,\mathbb R)$, pick a smooth form $\tau$ in the same class as $[T]$, then does the wedge map $\tau^q\wedge :H^0(X,\Omega^{n-q})\to H^q(X,\Omega^n)$ induces a surjective map?

This theorem is already known for the Kähler case, for a compact Kähler manifold $X$, replace here $\tau$ by a Kähler form $\omega$, then the wedge map $\omega^q\wedge:H^0(X,\Omega^{n-q})\to H^q(X,\Omega^n)$ induces a surjective map. What if we generalize the Kähler case to Fujiki class $\mathcal C$？ do we have a similar conclusion as stated above? Does anyone knows any reference about this problem?

Added:from Ang14, page7 theorem 0.10, we know

For a compact manifold $X$ endowed with a symplectic structure $\omega$, $X$ satisfies the hard Lefschetz conditon and $X$ satisfies the $dd^{\wedge}$-lemma (namely, every $d$-exact $d^{\wedge}$-closed form being $dd^{\wedge}$-exact) are equivalent.

see the same page for the defintion of $d^{\wedge}$. So this provides a special case of our question, that is if we further assume $\tau$ being a symplectic form and the $dd^{\wedge}$-lemma is satisfied, the map $\tau^q\wedge :H^0(X,\Omega^{n-q})\to H^q(X,\Omega^n)$ is a surjective map. But our original problem remained open.