Non-extremality of horizons in the positive-curvature cosmological case

Let $k$ be the curvature of the symmetric surfaces, let $\Lambda$ be the cosmological constant, and consider generic initial data for the surface-symmetric Einstein-Vlasov system. Let

$$and$$

denote the two horizon alternatives defined earlier in the paper. Non-extremality conjecture. For generic initial data in the case $k=1$, $\Lambda>0$, all horizons satisfy

$$or$$

. A positive resolution would imply strong cosmic censorship in this case when the relevant earlier hypothesis holds, although the authors note that strong cosmic censorship might also be proved without this conjecture.