1) expected inflation is equal to the true expected value of inflation given the policy rule so it changes when the policy changes and not just because past inflation changes. If the models are taken literally, this happens instantly. Of course all assume that there is really some learning dynamics -- but that is enough to weaken the claim enough for it to be untestable given historical data. Now I don't think that anyone could doubt Volcker's resolve in 1982. Survey data (of experts even) from the trough of the ruthless Volcker recession show they expected inflation to increase. This can be explained if one assumes that Volcker hadn't demonstrated resolve or that the survey participants were distracted by flying unicorns. The second story is much more plausible.
2) desired prices are a markup on marginal costs. This requires managers to have a firm opinion about what their marginal costs are and to believe that it is a useful concept. To believe that, you have to act as if "as if" were a convincing response to any fact. Notoriously managers either are not familiar with the concept or say they have no use for it. Micro foundations means ignoring what actual economic agents say about what they do. I have no idea (and have long wondered) how anyone can claim to believe in microfoundations and also use "as if" arguments. This is a plain obvious blatant contradiction.
3) anticipated future marginal costs have to matter too. Brad asked "Thus your standard New Keynesian model will use Calvo pricing and model the current inflation rate as tightly coupled to the present value of expected future output gaps. Is this a requirement anyone really wants to put on the model intended to help us understand the world that actually exists out there? " He sure is insinuating nK models of price formation are nonsense, and I don't know of anyone answering the question.