So, why don't most CEOs regularly show up to work in their pajamas if social status is mostly a matter of interaction between people with different levels of status, and other signaling is "only the tip of the iceberg"? And if you say that the behavior of CEOs is an exception, and the behavior of celebrities is (somewhat) an exception due to expected behavior from celebrities, and the behavior of the P.O.T.U.S. and other politicians are exceptional -- whose behavior is explained well by this theory?

While interactions between people of different levels of status are important for an individual's social status, I suspect that it is not remotely the most important factor.