Isotriviality conjecture for families over polydisks with minimal fibers

Let $\pi\to X\to \Delta^m$ and $S$ satisfy the property $(P)$. A projective manifold $S$ is minimal if its canonical line bundle $\omega_S$ is nef.

Isotriviality conjecture for minimal manifolds. If $S$ is minimal with $\dim S\geq 4$, then every fiber of $\pi$ is biholomorphic to $S$.

This conjecture is presented as a consequence of the deformation-rigidity theorem and as a statement that appears easier to establish than the abundance conjecture. The notation and property $(P)$ are paper-specific, so their precise content should be checked in the source.