Strong cosmic censorship for spherical cosmological spacetimes with positive cosmological constant

Let $k$ be the curvature of the symmetric surfaces and let $\Lambda$ be the cosmological constant. Consider surface-symmetric cosmological spacetimes with collisionless matter and generic initial data. Strong cosmic censorship conjecture. Strong cosmic censorship holds in the case $k=1$, $\Lambda>0$. This is identified in the paper as the first obvious open problem left unresolved; the remaining obstruction concerns possible extremal Schwarzschild-de Sitter-type black holes.