@CharlesRezk Good question. It is certainly known that you get a spectrum with homotopy groups: ?? Z/2 Z/2 C^times and that the top part is essentially a chunk of the Brown-Comenetz dual of the sphere (using C^\times in place of Q/Z). I am fairly certain that the bottom homotopy group (??) is trivial, just as in the non-super case. In fact I think the same argument works using a super version of global dimension, but I am not 100% sure. The literature on super tensor categories is less extensive.