Closedness conjecture for balanced manifolds

Let $\pi:\mathcal{X}\to\Delta$ be a holomorphic family of compact complex manifolds, with fibres $X_t$. A compact complex manifold is balanced if it has the balanced property. Balanced closedness conjecture. The balanced property of compact complex manifolds is closed under holomorphic deformations.

The balanced property is not deformation open, as shown by Alessandrini and Bassanelli, so this conjecture concerns the opposite stability direction. The paper presents it as an expected analogue of the strongly Gauduchon closedness conjecture; no resolution is given.