The rigidity conjecture for automorphism groups of affine varieties

Let $\mathbb K$ be an algebraically closed field of characteristic zero. Let $Y$ be an affine algebraic variety. Call $Y$ rigid if it admits no effective $\mathbb G_{\rm a}$-action, equivalently if $\operatorname{Aut}^{0}(Y)$ contains no $\mathbb G_{\rm a}$-subgroup. Rigidity conjecture. If $Y$ is rigid, then $\operatorname{Aut}^{0}(Y)$ is an algebraic torus of rank at most $\dim Y$. The conjecture predicts that the identity component of the automorphism group of a rigid affine variety is algebraic and has the expected dimension bound; the paper's abstract states this as the phenomenon established for affine surfaces, while the displayed conjecture concerns arbitrary affine varieties.