Outermost apparent horizon conjecture for black holes

Let

$$be initial data as in Theorem 1, with maximal future development, quotient spacetime$$

, black-hole region, apparent horizon

$$, and outermost apparent horizon$$

. Assume the black hole is sub-extremal in the limit and

$$. **Outermost apparent horizon conjecture.** There \exists a nonempty asymptotically connected component$$

of the outermost apparent horizon that terminates at $i^+$. Moreover, in a sufficiently small neighborhood

of $i^+$,

\mathcal{A}'\cap\mathcal{U}=\mathcal{A}\cap\mathcal{U},

$$and$$

I^+(\mathcal{A}\cap\mathcal{U})\cap\mathcal{Q}^+\subset\mathcal{T}.

The conjecture describes the geometry of the trapped region near $i^+$ and, according to the paper, would imply continuous extension of $r$ to

near $i^+$. It remains unresolved in the supplied source.