Automorphism groups of rigid affine varieties

Prove that if X is a rigid affine variety, then the identity component of its automorphism group Aut(X) is an algebraic torus.

Background

A rigid affine variety is an affine variety admitting no nontrivial action of the additive group. The paper explains that, except for the affine line, a non-rigid affine variety has an infinite-dimensional identity component of its automorphism group. The conjecture therefore asserts that, for rigid affine varieties, an algebraic torus is the only possible identity component of the automorphism group.

The conjecture is cited as having been verified in several special settings, including surfaces, toric varieties, and complexity-one T-varieties, but it is not established in general. The paper’s main theorem concerns a different conjectural generalization and does not resolve this statement universally.