The rigid-affine trivial-automorphism conjecture

Let $X$ be an affine variety admitting no effective $\mathbb G_{\mathrm a}$-action and no effective $\mathbb G_{\mathrm m}$-action. The rigid-affine trivial-automorphism conjecture. The identity component $\operatorname{Aut}^0(X)$ is trivial. This is presented as a weaker form of the rigid-affine automorphism-torus conjecture; the source gives no resolution.