Conductors in p-adic families

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Given a Weil-Deligne representation of the Weil group of an ℓ-adic number field with coefficients in a domain O, we show that its pure specializations have the same conductor. More generally, we prove that the conductors of a collection of pure representations are equal if they lift to Weil-Deligne representations over domains containing O and the traces of these lifts are parametrized by a pseudorepresentation over O.