Horizon condition for spherically symmetric asymptotically flat Einstein-Vlasov data

Let $\mathcal{H}^+$ be the future event horizon and let

$$denote the horizon condition defined earlier in the paper. Consider all spherically symmetric asymptotically flat initial data for the Einstein-Vlasov system. **Horizon-condition conjecture.** For all spherically symmetric asymptotically flat initial data,$$

is satisfied on $\mathcal{H}^+$. This is stated as a stronger conjecture implying the absence of future Cauchy horizons; the authors note that proving it only for generic initial data would be sufficient for applications to strong cosmic censorship.