I had my own debatable models in mind. One was the "ergodic plus slow energy loss" model. In this model, all orientation and rotation-axis (uniform on sphere) states are equally likely at any time, and energy (rotational kinetic plus gravitational potential) is assumed to be lost gradually and very slowly. At some moment it will happen that the dice becomes trapped in a local energy basin which it can thereafter never escape from because the energy keeps decreasing after the moment energy falls below the barrier height. Then only orientations within the current basin are permitted (uniform within) -- well more generally we only permit orientations below the current energy. So then there is a "landscape" of gravitational energy as function of orientation, and the basin-measures correspond to the probabilities we are seeking. It actually is not as nasty as all that may sound because for a convex-polyhedron-die, the barrier heights are easy to work out (they arise from polyhedron edges...) and the basins correspond to certain polygonal regions drawn on the surface of the sphere of orientations using circular arcs as polygon edges. In full generality life could be complicated, but for a symmetrical pyramid die, which was what I originally had in mind, it appears it really is within reach to work it out in full in this model.