Discreteness conjecture for automorphism groups without additive and multiplicative actions

Let $Y$ be an affine variety over an algebraically closed field of characteristic zero. Assume that $Y$ admits no effective action of the additive group $\mathbb G_{\rm a}$ and no effective action of the multiplicative group $\mathbb G_m$. Discreteness conjecture. Then $\operatorname{Aut}(Y)$ is a discrete group. This statement is presented as a consequence of the rigidity conjecture, and is attributed in the source to a suggestion by Hanspeter Kraft. Its status therefore depends on the first conjecture and is not resolved in the supplied text.