As we know, a compact Kähler manifold remains Kähler after any infinitesimal deformations. Since a compact complex manifold in Fujiki class $\mathcal C$ is bimeromorphic to a Kähler manifold, it was conjectured that Fujiki class $\mathcal C$ is also in class $\mathcal C$ of Fujiki after a small deformation, but later, Campana91 and LP92 proved that it was wrong, the class $\mathcal C$ of Fujiki is not stable by small deformations. The counterexample even exists in dimension 3, and in dimension $\geq 4$ even a Moishezon manifold has a small deformation which is not in Fujiki class $\mathcal C$.
Here, my question is, if a compact complex manifold in class $\mathcal C$ of Fujiki with canonical bundle trivial, then after a small deformation, is it still in class $\mathcal C$ of Fujiki? Or any other conditions which make the small deformations of Fujiki class $\mathcal C$ still in class $\mathcal C$? for example, admits a symplectic structure?
