Convex spacelike hypersurface of constant curvature with boundary on a hyperboloid

We consider convex, spacelike hypersurfaces with boundaries on some hyperboloid (or lightcone) in the Minkowski space. If the hypersurface has constant higher order mean curvature, and the angle between the normal vectors of the hypersurface and the hyperboloid (or the lightcone) is constant on the boundary, then the hypersurface must be a part of another hyperboloid.