Semisimplicity or virtual unipotence conjecture for automorphisms of compact Kähler threefolds

Let $X$ be a compact Kähler threefold and let $f\colon X\to X$ be an automorphism. The induced action

$$f^*_2\colon H^2(X,\mathbb R)\to H^2(X,\mathbb R)$$

Semisimplicity or virtual unipotence conjecture. The action $f^*_2$ is either semisimple or virtually unipotent. The mixed case is not possible for automorphisms of compact Kähler surfaces, while in dimension three the paper derives constraints but does not exclude it; no examples are known.