Karlsson–Nussbaum boundary-limit conjecture for fixed-point-free maps

Let $(\Omega,\delta)$ be a Hilbert geometry, let $f\colon\Omega\to\Omega$ be a fixed point free non-expansive mapping, and let $\omega(x,f)$ denote the set of limit points of the forward orbit of $x$. A subset of $\partial\Omega$ is convex when it contains the line segment between any two of its points.

Karlsson–Nussbaum conjecture. There exists a convex set $\Lambda$ in $\partial\Omega$ such that

$$\omega(x,f)\subseteq\Lambda\quad\text{for all }x\in\Omega.$$

This conjecture would constrain all orbit-limit sets of a fixed-point-free non-expansive map to a single convex subset of the boundary. The supplied text gives a counterexample showing that stronger statements about prescribed boundary limit sets fail, but does not report a resolution of this conjecture.