Karlsson–Nussbaum conjecture for finite-dimensional Hilbert metric spaces

Let $\Sigma$ be a convex set whose Hilbert metric is $\delta$, and let $f\colon\Sigma\to\Sigma$ be a fixed-point-free mapping. For $x\in\Sigma$, write

$$\mathcal{O}(x;f):=\{f^k(x)\colon k\geq 0\}.$$

Karlsson–Nussbaum conjecture. If $(\Sigma,\delta)$ is finite-dimensional, then there exists a convex set $\Omega\subseteq\partial\Sigma$ such that, for every $x\in\Sigma$, all accumulation points of $\mathcal{O}(x;f)$ lie in $\Omega$.

This conjecture is a Denjoy–Wolff-type assertion for nonexpansive mappings of finite-dimensional Hilbert metric spaces. The supplied text identifies it as a conjecture of Karlsson and Nussbaum and does not state that it has been resolved.