If this happened to you, please wait for a new game day, then check with Postal Bear. There is a catch-up mechanic that should mail you another signet ring that you can turn in to Emperor Bear.
(As of version 2.3.0 the game should be smart enough not to give you both quests simultaneously. However, it could still be an issue if one or both quests were set in motion prior to version 2.3.0.)
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