Earman's hole-free cosmic censorship conjecture

Let $(M,g_{ab})$ be an inextendible, hole-free, vacuum solution. A spacetime is strongly causal if every point $p$ and neighborhood $O$ of $p$ have a neighborhood $V\subseteq O$ that no causal curve intersects more than once. Let $H^+(S)$ denote the future Cauchy horizon of a slice $S$. Earman's hole-free cosmic censorship conjecture. If $S\subset M$ is a slice and there exists a point $p\in H^+(S)$, then either strong causality is violated at $p$ or

$$\overline{I^-(p)\cap S}$$

is noncompact. The conjecture gives a formulation of cosmic censorship for inextendible, hole-free vacuum spacetimes; the source states that it remains open in four-dimensional spacetime.