Perepechko–Zaidenberg conjecture on automorphisms of rigid affine varieties

Let $Y$ be a rigid affine algebraic variety over ${\mathbb K}$. The connected component $\operatorname{Aut}^0(Y)$ is an algebraic torus of rank at most $\dim Y$. Perepechko–Zaidenberg conjecture. If $Y$ is a rigid affine algebraic variety over ${\mathbb K}$, then the connected component $\operatorname{Aut}^0(Y)$ is an algebraic torus of the rank not greater than $\dim Y$. The paper notes that this conjecture holds for toral varieties over uncountable fields; its general status is not established here.