Tits-alternative conjecture for finitely generated unipotent subgroups

Let $X$ be an affine variety of dimension at least $2$, and let

$$G=\langle H_1,\ldots,H_k\rangle\subseteq\operatorname{Aut}(X)$$

be generated by finitely many ${\mathbb G}_a$-subgroups. Suppose that $G$ acts $2$-transitively on one of its orbits. The Tits-alternative conjecture asserts that $G$ contains a non-commutative free subgroup. The question is motivated by the known dichotomy for groups generated by root subgroups of affine toric varieties, while the stated implication remains open.