Let $G$ be a connected, simply connected, solvable, complex Lie group with a discrete subgroup $\Gamma$. Let also $G_a$ be Hochshild-Mostow hull of $G$, i.e., there exists a solvable linear algebraic group $G_a =({\mathbb C}^*)^k \ltimes G$ such that $G_a$ contains $G$ as a Zariski dense, topologically closed, normal complex subgroup.

Is it true that algebraic closure of $\Gamma$ and $G$ in $G_a$ are the same?