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We consider communication over a time invariant discrete memoryless channel with noiseless and instantaneous feedback. We assume that the communicating parties are not aware of the underlying channel, however they know that it belongs to some specific family of discrete memoryless channels. Recent results (A. Tchamkerten and I.E. Telatar) show that for certain families (e.g., binary symmetric channels and Z channels) there exists coding schemes that universally achieve any rate below capacity while attaining Burnashev's error exponent. We show that this is not the case in general by deriving an upper bound to the universally achievable error exponent.