Deligne's absolute Hodge-tensor conjecture

Let $({\mathbb V}^{\sigma})_{\sigma}$ be a de Rham motivic variation of Hodge structure on a smooth connected complex quasi-projective variety $S$. Let ${\mathbf G}$ be the group defined by the Hodge tensors and ${\mathbf G}^{\mathrm{AH}}$ the group defined by the absolute Hodge tensors. Deligne's conjecture. All Hodge tensors are absolute Hodge tensors, equivalently

$${\mathbf G}={\mathbf G}^{\mathrm{AH}}.$$

In the geometric case this is motivated by the behavior of algebraic cycles under automorphisms of ${\mathbb C}$; it remains open in the general motivic-variation setting.