Real-model quotient conjecture for commutative group varieties

Let $G$ be a commutative group variety over $\overline{\mathbb{Q}}$, and let $C$ be a connected proper real-analytic submanifold of $G(\mathbb{C})$. Suppose $C$ contains a subgroup $\Gamma\subset G(\overline{\mathbb{Q}})$ of algebraic points that is dense in $C$ for the analytic topology, and suppose every nontrivial $\xi\in\Gamma$ generates a Zariski-dense subgroup of $G(\mathbb{C})$. Real-model quotient conjecture. There exist a positive-dimensional algebraic group $G'$ over $K=\overline{\mathbb{Q}}\cap\mathbb{R}$ and a surjective algebraic homomorphism

$$v:G\longrightarrow G'\otimes_K\overline{\mathbb{Q}}$$

with $v(C)=G'(\mathbb{R})$. This generalizes the abelian-variety conjecture by replacing an isogeny with a surjective homomorphism. The source motivates it as a conjectural substitute after exhibiting an obstruction to the direct generalization, but gives no resolution.