Finiteness question for automorphism groups of toral varieties

Let $Y$ be a toral variety, and suppose that its maximal torus in $\operatorname{Aut}(Y)$ is trivial. Finiteness question. Is $\operatorname{Aut}(Y)$ a finite group? This is posed after proving finiteness under the additional hypothesis $\operatorname{rk} E(Y)=\dim Y+1$. The source does not state whether the question has been resolved.