The zero-entropy conjecture for wild automorphisms of compact Kähler spaces

Let $X$ be a compact Kähler space, and let $\sigma\in\operatorname{Aut}(X)$ be a wild automorphism. Zero-entropy conjecture. The automorphism $\sigma$ has zero entropy. The statement is known for compact complex surfaces and, by the results in this paper, for compact Kähler threefolds; it remains open in general.