Nussbaum–Vandehey conjecture on Hilbert nonexpansive maps

Let $C$ be a bounded convex domain in $\mathbb{R\mathrm{^{N}}}$ equipped with Hilbert's metric $d$, and let $f:C\rightarrow C$ be a 1-Lipschitz map. Nussbaum–Vandehey conjecture. Either $f$ has a fixed point in $C$, or the forward orbits $f^{n}(x)$ accumulate at one unique closed face. This conjecture concerns the asymptotic behavior of nonexpansive self-maps of Hilbert geometries; the supplied text identifies it as having been made independently by Nussbaum and the author, but gives no evidence of resolution.