Future causal geodesic completeness conjecture for symmetric cosmological spacetimes

Let $(\mathcal{M},g)$ be the maximal Cauchy development of $T^2$-symmetric or $k=-1$ surface-symmetric initial data in the vacuum or with Vlasov matter and with $\Lambda \ge 0$. Assume $(\mathcal{M},g)$ is non-flat. Denote by $t$ the area of the orbits of symmetry and orient $(\mathcal{M},g)$ by $\nabla t$. Future causal completeness conjecture. Then $(\mathcal{M},g)$ is future causally complete. This conjecture extends known future geodesic-completeness results for Gowdy spacetimes, for small data in the $k=-1$ surface-symmetric case, and for $\Lambda>0$. The general case for non-flat $T^2$-symmetric and $k=-1$ surface-symmetric spacetimes remains open.