Let $\mathcal X\to B$ be a holomorphic family of compact Kähler manifolds, let $\mathcal K_t$ denote the Kähler cone of the fiber $X_t$, then the union $\cup_{t\in B}\mathcal K_t$ forms an open set in the Hodge bundle $\mathcal H^{1,1}_{\mathbb R}$ (See for example, Voisin's paper, p.26 paragraph 4, which implies that the union of Kähler cones form an open set in $\mathcal H^{1,1}_{\mathbb R}:=\cup_{t\in B}H^{1,1}(X_t,\mathbb C)$), where $\mathcal H^{1,1}_{\mathbb R}$ is the subbundle of the bundle $\mathcal H^k:=R^k\pi_*\mathbb C\otimes\mathcal O$ with the fiber $H^{1,1}(X_t,\mathbb R)$ over $t\in B$.
Similarly, for a family of manifolds in Fujiki class $\mathcal C$, we call the de Rham class $[T]$ of a Kähler current $T$ a big class, and we call the open convex cone $\mathcal F_t$ formed by big classes $[T_t]\in H^{1,1}(X_t,\mathbb R)$ a Fujiki cone, then the family of Fujiki cones form an open set in the Hodge bundle $\mathcal H^{1,1}_{\mathbb R}$? If not, what condition can make these cones form an open set in $\mathcal H^{1,1}_{\mathbb R}$?
Remark: the fujiki class $\mathcal C$ is not stabel under small deformations, see for example: The class is not stable by small deformations-Campana, so here we require all the fibers remain in Fujiki class $\mathcal C$.
