Baldi–Klingler–Ullmo's typical and atypical Hodge locus conjecture

Let $(\mathbb{V},\mathscr{F}^{\bullet})$ be an integral polarized variation of Hodge structures on a smooth connected quasi-projective variety $S$. The Baldi–Klingler–Ullmo conjecture. The typical Hodge locus for $(\mathbb{V},\mathscr{F}^{\bullet})$ is either empty or dense in $S^{\text{an}}$ for the complex analytic topology, and the atypical Hodge locus for $(\mathbb{V},\mathscr{F}^{\bullet})$ is algebraic, namely, the set of atypical special subvarieties of $S$ has finitely many maximal elements under inclusion. This conjecture describes the expected dichotomy between typical and atypical Hodge loci; the cited work proves related algebraicity results in certain cases, but the general assertions remain open.