Lang's conjecture for intersections with finite-rank subgroups

Let $A$ be a complex abelian variety, let $\Gamma$ be a subgroup of finite rank in $A(\mathbb{C})$, and let $X$ be a subvariety of $A$. Here $\overline{X(\mathbb{C}) \cap \Gamma}$ denotes the Zariski closure of $X(\mathbb{C}) \cap \Gamma$. Lang's conjecture. There are abelian subvarieties $C_1, \ldots, C_n$ of $A$ and elements $\gamma_1, \ldots, \gamma_n \in \Gamma$ such that

$$\overline{X(\mathbb{C}) \cap \Gamma} = \bigcup_{i=1}^n (C_i + \gamma_i)$$

and

$$X(\mathbb{C}) \cap \Gamma = \bigcup_{i=1}^n (C_i(\mathbb{C}) + \gamma_i) \cap \Gamma.$$

The statement is presented as the absolute form of Lang's conjecture and is substantially included in the theorem proved immediately before it; the source does not provide an independent status discussion for this formulation.