Given compact Kähler manifolds $X$ and $X'$ deformation equivalent over the unit disk $\Delta \subset \mathbb{C}$. More precisely, there is a proper holomorphic surjective map \begin{align*} \pi\colon \mathcal{X}\to \Delta \end{align*} and $t,t' \in \Delta$ such that $X$ and $X'$ are biholomorphic to the fibers $\pi^{-1}(t), \pi^{-1}(t')$ respectively. Is there a deformation of $X$ and $X'$ over $\Delta$ such that every fiber is Kähler?

I am specially interested in the case where $X$ and $X'$ are of hyperkähler type, i.e. irreducible holomorphic symplectic. In other words, simply connected and admitting a unique holomorphic symplectic form. I know that there are (large) deformations where the deformed space is not Kähler.