Non-homeomorphism conjecture for infinity harmonic maps

Let $u$ be a limit infinity harmonic map between hyperbolic surfaces, and let $\lambda$ be the canonical geodesic lamination with image $\mu$. Non-homeomorphism conjecture. The limit infinity harmonic maps are not homeomorphisms; small regions around the canonical geodesic lamination are mapped onto $\lambda$, and in particular $u^{-1}(\mu)$ often contains open sets. This is a qualitative conjecture about the structure of the limit maps and remains open.