Baldi–Klingler–Ullmo's density conjecture for typical Hodge loci

Let $\mathbb V$ be a polarizable $\mathbb Z$-VHS on a smooth, irreducible and quasi-projective variety $S$ over $\mathbb C$. The typical Hodge-locus density conjecture. If $\mathrm{HL}(S, \mathbb V^\otimes)_\mathrm{typ}$ is non-empty, then $\mathrm{HL}(S, \mathbb V^\otimes)_\mathrm{typ}$ is analytically, hence Zariski, dense in $S$. The conjecture concerns the expected density of typical intersections, in contrast with the atypical part governed by the Zilber–Pink conjecture; the paper develops sufficient criteria for this density.