It's interesting that you focus on the common knowledge assumption as the really strict assumption, rather than Bayesian-ness.

The common-knowledge condition really is surprisingly strong. I think that this is especially clear from the definition that I gave in my write-up. The common knowledge C is a piece of information so strong that, once you know it, your posterior probability for the proposition A is totally fixed — no additional information of any kind can make you more or less confident in A.