Perepechko–Regeta conjecture on additive actions and nested automorphism groups

Let $X$ be an affine variety over an algebraically closed field ${\mathbb K}$ of characteristic zero. Let ${\mathcal U}(X)\subset \operatorname{Aut}(X)$ be the subgroup generated by all ${\mathbb G}_a$-actions, and let $\operatorname{Aut}^\circ(X)$ be the neutral component of the automorphism group. The subgroup $\operatorname{Aut}^\circ(X)$ is nested when it is a nested ind-subgroup, that is, a direct limit of algebraic subgroups. Perepechko–Regeta conjecture. ${\mathcal U}(X)$ is abelian if and only if $\operatorname{Aut}^\circ(X)$ is nested. This is proposed as an extension of the paper's theorem for $\operatorname{Aut}_{\operatorname{alg}}(X)$; the extension remains open.