He tried to tell me that I didn't need to but I told him the battery drains precipitously in the cold and that'd I prefer we switch it and he just said ok and started the process.

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To answer your question, I guess it couldn't hurt? The thing is that I definitely did not trust the 81% that the battery health check was telling me. My battery would drain in minutes from 30% and the phone was practically useless in the cold.