Rigidity conjecture for families with decomposable transcendental Hodge structures

Let $\pi: \mathscr{S} \to B$ be a family of surfaces over an analytic space $B$, and call a point of $B$ very general if it lies outside a countable union of proper analytic subsets. A family is isotrivial when all its fibers are isomorphic, equivalently when it is locally trivial for the étale topology. Rigidity conjecture. If a very general fiber $\mathscr{S}_b$ has a decomposable integral polarized Hodge structure on its transcendental lattice, then the family

$$\pi: \mathscr{S} \to B$$

is isotrivial. This conjecture asserts that decomposability of the transcendental Hodge structure cannot occur generically in a genuinely varying family of surfaces. Its status is not resolved in the supplied text.