BKL conjecture is named after V.A.Belinski, I.M. Khalatnikov and E.M. Lifshitz, three russian physicists, that proposed it in BeKhLi1970. This conjecture describes the behavior of a gravitational field approaching a singularity.
The conjecture can be formulated by saying that Einstein equations become local near a singularity and the solution becomes oscillatory. The first part of the conjecture claims that the spatial part of the Einstein equations is negligible approaching a singularity. Indeed, this can be proven with methods given in perturbation theory and this was done in FraA2006. A numerical proof was also given recently Garf2004. Oscillatory behavior is also obtained.
BKL conjecture implies a weak formulation of the Cosmic Censorship Hypothesis as the spacetime near a singularity is spacelike in this case.