To my mind the latter is logically impossible. Subjective self-continuity is not intrinsic to the experience of consciousness, it is a conclusion we draw upon introspection, based on circumstantial evidence. Subjectively, all we ever have is a sequence of fleeting moments. We only have the "word" of our memory to trust that the Now Me is a descendant of Past Me. As soon as perfect replication is possible, the replicant and the original have exactly equivalent claims to identity in every sense that matters, just one will have a better claim to physical continuity. But physical continuity is not the most important aspect of identity, it's just our intuitions favor it so heavily because it has historically perfectly coincided with conscious identity.