Lie-algebra closure conjecture for finitely generated unipotent subgroups

Let $X$ be an irreducible affine variety. For locally nilpotent derivations $\partial_1,\ldots,\partial_k$ of ${\mathbb K}[X]$, let

$$H_i={\rm \exp}({\mathbb K}\partial_i),\qquad G=\langle H_1,\ldots,H_k\rangle\subseteq\operatorname{Aut}(X).$$

A Lie-algebra closure conjecture asserts that a ${\mathbb G}_a$-subgroup $H={\rm \exp}({\mathbb K}\partial)$ lies in $\overline{G}$ if and only if $\partial$ belongs to the Lie algebra generated by $\partial_1,\ldots,\partial_k$. This is proposed as a generalization of a lemma for root subgroups; the supplied text gives no resolution status.