Lie-algebraic closure conjecture for finitely generated additive subgroups

Let $X$ be an affine variety and let $A=\mathcal{O}_X(X)$ be its structure algebra. For locally nilpotent derivations $\partial_1,\ldots,\partial_k\in\operatorname{LND}(A)$, set

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

For a locally nilpotent derivation $\partial\in\operatorname{LND}(A)$, let $H=\mathop{\rm \exp}(\mathbb{K}\partial)\subset\operatorname{SAut}(X)$. Lie-algebraic closure conjecture. The subgroup $H$ is contained in $\overline{G}$ if and only if

$$\partial\in\operatorname{Lie}\,\langle\partial_1,\ldots,\partial_k\rangle.$$

The conjecture concerns when the closure of a subgroup generated by finitely many additive one-parameter subgroups contains another such subgroup. It arises after a theorem characterizing algebraicity of generated groups through finite-dimensionality of the Lie algebra generated by their Lie algebras; its status is not resolved in the supplied text.