Type A characterization by LND images and stable points

Let $Y$ be a variety, let ${\mathbb K}[Y]$ be its coordinate ring, and let $I$ be the ideal of ${\mathbb K}[Y]$ generated by the images of all locally nilpotent derivations on $Y$. Let $\mathrm{SAut}(Y)$ denote the subgroup of $\operatorname{Aut}(Y)$ generated by additive-group actions, and call a point $\mathrm{SAut}(Y)$-stable if it is stable under this group. Type A conjecture. The following conditions are equivalent:

• $Y$ is of type A;
• $I={\mathbb K}[Y]$;
• $Y$ has no $\mathrm{SAut}(Y)$-stable points.

The conjecture proposes that the two existing descriptions of type A varieties—via the modified Derksen invariant and via the absence of stable points—are equivalent, beyond the cases established by the preceding propositions.