Impose symmetric psd with minimum eigenvalue, or symmetric nsd with maximum eigenvalue, either of which can be done in CVX. I guess you were already down this road with your previous topic How to write this in CVX?
Otherwise, I think you’re out of luck, and even as a non-convex problem, it could be a nightmare from a numerical (conditioning) standpoint.
Maybe you need to rethink the math of whatever it is you are really trying to do in the big picture. This forum is probably not the best place for that, as a general prerequisite is that you come with a reasonably well-specified (possibly mixed-integer) convex optimization problem, which you do not have.