Abate's weak Wolff–Denjoy conjecture for taut acyclic manifolds

Let $\mathscr{X}$ be a taut, acyclic complex manifold, meaning that $H_j(\mathscr{X},\mathbb{Z})=0$ for all $j\in\mathbb{N}$, and let $f\in\operatorname{Hol}(\mathscr{X},\mathscr{X})$. The iterates of $f$ are denoted by $\{f^k\}$. Abate's weak Wolff–Denjoy conjecture. The sequence $\{f^k\}$ is compactly divergent if and only if $f$ has no fixed point. This conjecture was disproved by Abate and Heinz in 1992, so the asserted equivalence is false in general.