Conjecture on density of the typical Hodge locus

Let ${\mathbb V}$ be a polarizable $\mathbb{Z}$-variation of Hodge structures on a smooth connected complex quasi-projective variety $S$. The typical Hodge locus $\operatorname{HL}(S, {\mathbb V}^{\otimes})_{\operatorname{typ}}$ is the union of the strict typical special subvarieties of $S$ for ${\mathbb V}$.

Density conjecture for the typical Hodge locus. If $\operatorname{HL}(S, {\mathbb V}^{\otimes})_{\operatorname{typ}}$ is nonempty, then it is dense in $S$ for the analytic topology.

Together with the strong Zilber–Pink conjecture, this predicts a dichotomy between an algebraic Hodge locus and an analytically dense one. The source gives no resolution status.