Density threshold for typical Hodge loci in the Torelli locus

Let $g\geqslant4$ be an integer, let $S\subset\mathcal A_g$ be a closed Hodge generic subvariety of dimension $q$, and let $\mathbb V$ be the induced polarizable $\mathbb Z$-VHS on $S$. The Torelli-locus typical-density conjecture. The typical Hodge locus of $S$ for $\mathbb V$ is analytically dense if and only if $q\geqslant g-1$. When this holds, the Hecke translates of $\mathcal A_{g-1}\times\mathcal A_1$ in $\mathcal A_g$ intersect $S$ in an analytically dense subset. This predicts the precise dimension threshold for density in the Siegel modular variety and identifies the largest relevant strict special subvarieties.