Zaidenberg's torus characterization of affine varieties without additive actions

Let $X$ be an affine variety over an algebraically closed field ${\mathbb K}$ of characteristic zero. Let ${\mathbb G}_a$ denote the additive group of ${\mathbb K}$, and let $\operatorname{Aut}^\circ(X)$ be the neutral component of the automorphism group of $X$. The variety $X$ admits no additive group actions when it has no ${\mathbb G}_a$-actions. Zaidenberg's conjecture. $X$ admits no additive group actions if and only if $\operatorname{Aut}^\circ(X)$ is an algebraic torus. The torus direction is known when $\operatorname{Aut}^\circ(X)$ is finite-dimensional and for certain $T$-varieties, but the stated equivalence is not resolved in general.