It is known that small deformations of a manifold $X$ in Fujiki class $\mathcal C$ might no longer be in $\mathcal C$, see This paper for example.

However, my question is:

For any $X$ in $\mathcal C$, does there always exists a deformation $\pi:\mathcal X\to B$ of $X$, such that for any small neighborhood of $X$, there exists a fiber $X_t:=\pi^{-1}(t)$ in it which is Kähler?

Does anyone has any counterexample?

It is known that small deformations of a manifold $X$ in Fujiki class $\mathcal C$ might no longer be in $\mathcal C$, see This paper for example.

However, the following question is still open:

For any $X$ in $\mathcal C$, does there always exists a deformation $\pi:\mathcal X\to B$ of $X$, such that for any small neighborhood $U$ of $X$, there exists a Kähler deformation $X_t:=\pi^{-1}(t)$ with $t\in U$?

Does anyone has any counterexample?

It is known that small deformations of a manifold $X$ in Fujiki class $\mathcal C$ might no longer be in $\mathcal C$, see This paper for example.

However, my question is:

For any $X$ in $\mathcal C$, does there always exists a deformation $\pi:\mathcal X\to B$ of $X$, such that for any small neighborhood of $X$, there exists a fiber $X_t:=\pi^{-1}(t)$ which is Kähler?

Does anyone has any counterexample?

It is known that small deformations of a manifold $X$ in Fujiki class $\mathcal C$ might no longer be in $\mathcal C$, see This paper for example.

However, my question is:

For any $X$ in $\mathcal C$, does there always exists a deformation $\pi:\mathcal X\to B$ of $X$, such that for any small neighborhood of $X$, there exists a fiber $X_t:=\pi^{-1}(t)$ in it which is Kähler?

Does anyone has any counterexample?