The dR-absolute specialness conjecture

Let $(\mathbb{V}^{\sigma})_{\sigma}$ be an absolute variation of Hodge structure on a smooth irreducible quasi-projective complex algebraic variety $S$. For a closed irreducible subvariety $Z\subset S$, let $G_Z$ be its generic Mumford–Tate group and let $G_Z^{AH}$ be its generic dR-absolute Mumford–Tate group; call $Z$ dR-absolutely special if it is maximal among closed irreducible algebraic subvarieties having generic dR-absolute Mumford–Tate group $G_Z^{AH}$. The dR-absolute specialness conjecture. If $Z\subset S$ is a special subvariety, then $Z$ is dR-absolutely special. This is weaker than Deligne’s conjecture that all Hodge cycles are dR-absolute Hodge, but it would still imply important arithmetic properties such as the definability of special subvarieties over $\bar{\mathbb{Q}}$; it is proved in the paper for weakly non-factor special subvarieties.