Reichstein–Rogalski–Zhang conjecture on wild automorphisms

Let $X$ be a projective variety, and let a wild automorphism $b[?m\sigmab[?m$ of $X$ be an automorphism such that whenever a non-empty Zariski-closed subset $Z$ satisfies $\sigma(Z)=Z$, one has $Z=X$.

Reichstein–Rogalski–Zhang conjecture. If $X$ admits a wild automorphism, then $X$ is isomorphic to an abelian variety.

This conjecture is known when $\dim X\leq 2$ and was proved in this paper when $\dim X\leq 3$ provided $X$ is not a Calabi–Yau threefold; the general case remains open.