Popovici's conjecture on limits of Kähler manifolds

Let $X$ be a compact complex manifold that is a limit of Kähler manifolds, meaning that for some family of deformations $\mathcal X\to T$ with $0\in T$ corresponding to $X$, there is a sequence of points $t_n\in T$ converging to $0$ such that each deformation $X_{t_n}$ is Kähler.

Popovici's conjecture. A limit of Kähler manifolds is of Fujiki class $\mathcal C$, that is, it is bimeromorphic to a Kähler manifold.

This conjecture concerns the stability of Fujiki class $\mathcal C$ under limits of Kähler manifolds and is relevant to the study of degenerations in complex geometry. The source presents it as expected, and does not state a resolution.