Achieving SK capacity in the source model: When must all terminals talk?

In this paper, we address the problem of characterizing the instances of the multiterminal source model of Csiszár and Narayan in which communication from all terminals is needed for establishing a secret key of maximum rate. We give an information-theoretic sufficient condition for identifying such instances. We believe that our sufficient condition is in fact an exact characterization, but we are only able to prove this in the case of the three-terminal source model.