Klingler's geometric André–Oort conjecture for variations of Hodge structures

Let ${\mathbb V}$ be a ${\mathbb Z}$-variation of Hodge structure on a smooth irreducible complex quasi-projective variety $S$. A special subvariety for $(S,{\mathbb V})$ is positive dimensional and of Shimura type with dominant period map when it has the properties specified in the source. Klingler's geometric André–Oort conjecture. If the set of positive-dimensional special subvarieties for $(S,{\mathbb V})$ that are of Shimura type with dominant period maps is Zariski-dense in $S$, then $(S,{\mathbb V})$ is of Shimura type with dominant period map. This is the geometric part of the preceding Shimura-type conjecture, concerning the distribution of positive-dimensional special subvarieties; the source gives no resolution status.