What does strong subadditivity tell us about black holes?

It has been argued that small corrections to evolution arising from non-geometric effects can resolve the information paradox. We can get such effects, for example, from subleading saddle points in the Euclidean path integral. But an inequality derived in 2009 using strong sub-additivity showed that such corrections cannot solve the problem. As a result we sharpen the original Hawking puzzle: we must either have (A) new (nonlocal) physics or (B) construct hair at the horizon.