Manifold $X$ in Fujiki class $\mathcal C$ with $c_1(X)=0$ admits arbitrarily small deformations which are Moishezon?

Question

It is known that any Calabi-Yau manifold $X$, i.e. compact Kähler manifold with $c_1(X)=0$, has arbitrarily small deformations which are algebraic (see, for example, Buchdahl's paper, Proposition 5).

Note that manifolds in Fujiki class $\mathcal C$ are bimeromorphic to compact Kähler manifolds, and Moishezon manifolds are bimeromorphic to projective manifolds, so do we have:

Any manifold $X$ in Fujiki class $\mathcal C$ with $c_1(X)=0$ admits arbitrarily small deformations which are Moishezon manifolds?

Answer 1

My answer is well overdue and incomplete but hopefully might be useful to you. According to the paper Moishezon deformations of manifolds in Fujiki class C the following analogue of Buchdahl's theorem holds:

Let $X$ be a manifold in Fujiki class with a Kähler current $T$, $\pi\colon \mathcal X\to B$ be a holomorphic family of manifolds in Fujiki class $\mathcal C$, $X = \pi^{-1}(0)$, suppose the union of Fujiki cones forms an open set in $\mathcal H^{1,1}_{\mathbb R}$ and the composition of the Kodaira-Spencer map: $T_B|_0\to H^1(X,T_X)$ with the natural map $\lrcorner [T]\colon H^1(X,T_X)\to H^{0,2}_{\bar\partial}(X)$ is surjective, then there exists an arbitrarily small deformation of $X$ which is a Moishezon manifold.

Local Torelli theorem holds for Calabi-Yau manifolds in Fujiki class $\mathcal C$ according to this paper. Hence we only need to check the first condition, i.e., that the union of Fujiki cones for all deformations is open. I have genuinely no idea how to check this and what can go wrong. And of course deformations might not a priori be in Fujiki class $\mathcal C$, but at least they satisfy $\partial\bar\partial$-lemma according to the second paper.