That has been proven for any non-extremal black hole in
Einstein gravity coupled to any matter obeying the weak
energy condition with hyperbolic equations of motion and asymptotically-flat boundary conditions [161, 163
, 254, 86, 143].
The proof has been extended to extremal black holes, to higher dimensions and to anti-de Sitter asymptotics
in [171, 170
, 85].