The Kähler wild automorphism conjecture for compact complex spaces

Let $X$ be a compact Kähler space. An automorphism $\sigma\in\operatorname{Aut}(X)$ is wild if every non-empty analytic subset $Z\subseteq X$ satisfying $\sigma(Z)=Z$ is equal to $X$. The Kähler wild automorphism conjecture. If $X$ admits a wild automorphism, then $X$ is isomorphic to a complex torus. This generalises the corresponding projective conjecture to the Kähler setting. It is proved for compact complex surfaces, while the general Kähler case remains open.