  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 ${\rm dim} \, X \le 2$) 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 ${\rm dim} \, X \le 3$
and $X$ is not a Calabi-Yau threefold, and also show that $\sigma$ is of zero
entropy when ${\rm dim} \, X \le 4$ and the Kodaira dimension $\kappa(X) \ge
0$.
