The non-torsion automorphism conjecture for infinite-dimensional automorphism groups

Let $X$ be a variety. The non-torsion automorphism conjecture. If $\operatorname{Aut}(X)$ is infinite dimensional (respectively, has a nontrivial connected component), then $\operatorname{Aut}(X)$ contains a subgroup isomorphic to either $\mathbb G_{\mathrm m}$ or $\mathbb G_{\mathrm a}$; equivalently, $\operatorname{Aut}(X)$ contains a non-torsion algebraic element. This matters because it predicts that positive-dimensional or infinite-dimensional automorphism phenomena arise from additive or multiplicative algebraic actions; the source gives no resolution.