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Fibring is a general mechanism for combining logics that provides valuable insight on designing and understanding complex logical systems. Mostly, the research on fibring has focused on its model and proof-theoretic aspects, and on transference results for relevant metalogical properties. Conservativity, however, a property that lies at the very heart of the original definition of fibring, has not deserved similar attention. In this paper, we provide the first full characterization of thefatcat:bzgwyfyyuvgmvcrexuvysrl6sq