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Recent work has extended this paradigm to speaker identity unlearning in zero-shot text-to-speech (ZS-TTS), the task of selectively erasing a model's ability to replicate a speaker's voice. Existing methods, however, quietly assume all unlearning requests arrive at once; an unrealistic assumption, since privacy-motivated removals arrive sequentially over time. We show this assumption breaks state-of-the-art methods: unlearning each new speaker fully revives previously unlearned speakers, reintroducing the very privacy risk unlearning was meant to eliminate. We present Cumulative ORThogonal Identity Suppression (CORTIS), the first framework for continual speaker identity unlearning in ZS-TTS that requires no access to previously-unlearned speaker data. 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