Lie-algebraic form of the generalized Mazur conjecture

Let $A$ be a simple abelian variety over $\overline{\mathbb{Q}}$, and let $\Gamma\subset A(\overline{\mathbb{Q}})$ be a subgroup of positive rank. Let $C$ be the closure of $\Gamma$ in $A(\mathbb{C})$ for the analytic topology, and let $C^o$ be the connected component of unity of $C$. Write $\mathfrak a=\operatorname{Lie}A$ and $K=\overline{\mathbb{Q}}\cap\mathbb{R}$. Lie-algebraic generalized Mazur conjecture. There exists a $K$-subspace $\mathfrak c\subset\mathfrak a$ such that

$$C^o=\operatorname{exp}_A(\mathfrak c\otimes_K\mathbb{R}).$$

The source states this as an equivalent formulation of the preceding conjecture, relating analytic closures to subspaces defined over the real algebraic field $K$. No resolution is supplied.