Stability conjecture for the Kerr-de Sitter and Kerr families

Let $\mathcal{B}_\Lambda$ denote either the subextremal Kerr-de Sitter family for $\Lambda>0$ or the subextremal Kerr family for $\Lambda=0$. Let $b_0\in\mathcal{B}_\Lambda$ be generated by initial data, and consider a small perturbation of those initial data for a solution of the Einstein vacuum equations.

Stability conjecture. The family $\mathcal{B}_\Lambda$ is a stable family of solutions to the Einstein vacuum equations with $\Lambda\geq 0$: the evolution of the small perturbation asymptotes in the appropriate sense to some $b_{\varepsilon}\in\mathcal{B}_\Lambda$.

Stability of these stationary black-hole families is a basic requirement for the final-state conjecture, which expects subextremal Kerr and Kerr-de Sitter spacetimes to describe physical final states. The supplied text does not specify whether this conjecture has been resolved.