The conjecture in question was stated in Margulis' 1990 ICM talk. In short, it says that for any partially hyperbolic homogeneous flow and any finite subset Z of the space, there are many orbits which are bounded and stay away from Z. Some partial results were obtained in the 1990s, but the full strength of the conjecture resisted any advances until recently. I will describe the plan of attack on the conjecture, which is work in progress joint with Barak Weiss.