Ringström's topology-obliviousness conjecture for expanding Einstein spacetimes

Let $(\widetilde{M}, \widehat{g})$ be a globally hyperbolic Lorentzian manifold with compact Cauchy hypersurfaces. Say that late-time observers are oblivious to topology if there exists a Cauchy hypersurface $M \subset \widetilde{M}$ such that no future-directed inextendible causal curve $\gamma$ satisfies $M \subset J^{-}(\gamma)$. Ringström's topology-obliviousness conjecture. If $(\widetilde{M}, \widehat{g})$ is a future causally geodesically complete solution of the vacuum Einstein equations with positive cosmological constant $\Lambda>0$ and compact Cauchy hypersurfaces, then late-time observers in $(\widetilde{M}, \widehat{g})$ are oblivious to topology. This conjecture concerns the causal inaccessibility of spatial topology at late times in expanding cosmological solutions; the supplied text identifies it as an open problem.