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 non-empty Zariski-closed subset $Z$ of $X$, we have $Z=X$. Then $X$ is conjectured to be an abelian variety with $\sigma$ of zero entropy (and proved to be so when $\mathrm{d}\mathrm{i}\mathrm{m}X\le 2$) by Z. Reichstein, D. Rogalski and J. J. Zhang in their study of projectively simple rings. This conjecture has been generally open for more than a decade. In this note, we confirm this original conjecture when $\mathrm{d}\mathrm{i}\mathrm{m}X\le 3$ and $X$ is not a Calabi-Yau threefold, and also show that $\sigma$ is of zero entropy when $\mathrm{d}\mathrm{i}\mathrm{m}X\le 4$ and the Kodaira dimension $\kappa(X) \ge 0$.