The Deligne absolute Hodge cycle conjecture

Let $Z\subset S$ be a closed irreducible subvariety of a variation of Hodge structure. Write $G_Z$ for its generic Mumford–Tate group and $G_Z^{AH}$ for its generic dR-absolute Mumford–Tate group. Deligne’s absolute Hodge cycle conjecture. All Hodge cycles are dR-absolute Hodge, equivalently,

$$G_Z=G_Z^{AH}.$$

Deligne’s conjecture is substantially stronger than the dR-absolute specialness conjecture and is known for families of abelian varieties, but remains open in general.