The discreteness conjecture for affine varieties without additive or multiplicative actions

Work over an algebraically closed field $\mathbb K$ of characteristic zero. Let $Y$ be an affine variety, and let $\mathbb{G}_{\rm a}$ and $\mathbb{G}_{\rm m}$ denote the additive and multiplicative algebraic groups. An action is effective if only the identity group element acts trivially. The discreteness conjecture. If $Y$ admits no effective $\mathbb{G}_{\rm a}$- or $\mathbb{G}_{\rm m}$-actions, then $\operatorname{Aut}(Y)$ is a discrete group. This is presented as a consequence of the preceding conjecture and is cited as an open problem in the source; it asserts that the absence of both types of one-parameter group action eliminates all positive-dimensional connected automorphisms.