A variant of the Mordell-Lang conjecture

The Mordell-Lang conjecture (proven by Faltings, Vojta and McQuillan) states that the intersection of a subvariety $V$ of a semiabelian variety $G$ defined over an algebraically closed field $\mathbb{k}$ of characteristic $0$ with a finite rank subgroup $\mathrm{\Gamma }\le G(\mathbb{k})$ is a finite union of cosets of subgroups of $\mathrm{\Gamma }$. We explore a variant of this conjecture when $G$ is a product of an abelian variety $A$ defined over $\mathbb{k}$ with the additive group ${\mathbb{G}}_a$.