Motivic Galois lifting conjecture for central torus coverings

Let $F$ and $E$ be number fields, and let $H'\twoheadrightarrow H$ be a surjection of linear algebraic groups over $E$ with central torus kernel. Suppose that

$$\rho\colon\mathcal{G}_{F,E}\to H$$

is a homomorphism. Motivic Galois lifting conjecture. If $F$ is imaginary, there should be a finite extension $E'/E$ and a homomorphism

$$\widetilde{\rho}\colon\mathcal{G}_{F,E'}\to H'_{E'}$$

lifting $\rho\otimes_EE'$. If $F$ is totally real, such a lift should exist if and only if the Hodge-number parity obstruction of Corollary fullmonodromytotreal vanishes. The conjecture generalizes Kuga–Satake-type lifting phenomena for motivic Galois representations and is presented as a broad conjectural framework; the source gives no resolution.