Rigidity for compact hyperbolic complex manifolds

We study the deformation behavior of compact hyperbolic complex manifolds. Let $\pi :\mathcal{X}\to \mathrm{\Delta }$ be a smooth family of compact complex manifolds over the unit disk in $\mathbb{C}$, and $H$ a compact hyperbolic complex manifold. Then the $H$-locus $\{t\in \mathrm{\Delta }:X_t\cong H\}$ is either at most a discrete subset of $\mathrm{\Delta }$ or the whole $\mathrm{\Delta }$. For a smooth family over a compact Riemann surface $Y$, its $H$-locus is either at most finite or the whole $Y$. Furthermore, if $Y$ is isomorphic to ${\mathbb{P}}^1$ or an elliptic curve, then we conjecture that the $H$-locus is empty or the whole $Y$.