  Consider a convex cone in three-dimensional Minkowski space which either
contains the lightcone or is contained in it. This work considers mean
curvature flow of a proper spacelike strictly mean convex disc in the cone
which is graphical with respect to its rays. Its boundary is required to have
constant intersection angle with the boundary of the cone. We prove that the
corresponding parabolic boundary value problem for the graph admits a solution
for all time which rescales to a self-similarly expanding solution.
