Wild automorphisms of projective varieties, the maps which have no invariant proper subsets

Let $X$ be a projective variety and $\sigma$ a wild automorphism on $X$, i.e., whenever $\sigma(Z) = Z$ for a Zariski-closed subset $Z$ of $X$, we have $Z = X$. Then $X$ is conjectured to be an abelian variety (and proved to be so when $\dim X \le 2$) by Z. Reichstein, D. Rogalski and J. J. Zhang. This conjecture has been generally open for more than a decade. In this note, we confirm this conjecture when $\text{dim} \, X \le 3$ and $X$ is not a Calabi-Yau threefold.