If the non-linear stability of the Kerr family is proven, then this will in particular produce examples of a generic family of black holes with regular horizon which dynamically asymptote to Kerr. A much more basic question, however, has up until now remained unanswered:
Do there exist any non-trivial examples of vacuum dynamic spacetimes which asymptote to Kerr?
The closest examples of such spacetimes previously known are the so-called Robinson–Trautman spacetimes. These, however, fail to be smooth on the horizon, and in any case, necessarily asymptote to Schwarzschild.
In recent joint work with Gustav Holzegel and Igor Rodnianski, see here, we have shown that given any parameters |a|≤M, there indeed exist dynamic vacuum spacetimes that settle down to the Kerr solution with parameters a and M, in fact, we can construct such examples with the full functional degree of freedom by prescribing "scattering data" for the Einstein vacuum equations on what will be the event horizon and null infinity, and solving backwards.