The absolute Hodge conjecture for generic Mumford–Tate groups

Let $S$ be the base of the family considered in the paper, and for a subvariety $Z\subset S$ let $G_Z$ and $G_Z^{AH}$ be its generic Mumford–Tate and generic absolute Mumford–Tate groups. Absolute Hodge conjecture. For any subvariety $Z\subset S$,

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

For a point this recovers the corresponding assertion for the fiber, while in families it asserts equality of the generic Hodge and absolute Hodge symmetry groups. The source presents this as a conjectural equality and does not state a resolution.