AdS instability conjecture for black-hole formation

Let $\operatorname{AdS}_{d+1}$ denote the Anti-de Sitter space in dimension $d+1$, with $d\geq 3$, and consider the spherically symmetric Einstein-scalar system with negative cosmological constant, Dirichlet asymptotics, and compatible smooth initial data as described above. AdS instability conjecture. The $\operatorname{AdS}_{d+1}$ space is unstable against the formation of a black hole for a large class of arbitrarily small perturbations. This conjecture concerns the nonlinear evolution of small perturbations in asymptotically Anti-de Sitter spacetime; the stated claim is supported by numerical investigations, while a general proof remains open.