Deep searches on relevant time scales (1-2 weeks) have revealed no such supernova-like signatures from short-durations GRBs (Figure 23).  In some cases, the limits are 7 magnitudes (1000x) deeper than the canonical GRB-associated supernova: SN1998bw.  This definitively rules out a supernova progenitor for short-duration GRBs.
But, if supernovae are not present in the late-time light curves of short-duration GRBs, is there another smoking gun signatures that unambiguously identify their progenitors?  We'll return to this question in our next lecture, when we introduce the concept of a kilonova.