I'd say it's different here though. Whatever happened to the other side of a (possibly entangled) bipartite state cannot affect your side, sure. In this case some information *is* passed between the two parties, so it seems conceivable that Bob might be able to predict what happens to his side better because of it (which corresponds to saying his description of his partial state should be updated).
I mean, it turns out this is not the case as per calculations above, but I'm not seeing why that should obviously be the case, yet