Interestingly, I think this is the answer that sank me. I spoke about dealing with, and resolving, conflict, finding common ground between people who are not getting along as a means to promote camaraderie, and finding ways to have people work on things that they enjoy. I don't think that is what they wanted (my management style was cited as the reason someone else got the offer - theirs was a "better fit"). I believe that they wanted someone to come in and clean house and/or be the sort of hardened manager that I cannot stand being around.