I too have the same problem. I've checked and cleared out every character's mailbox. Still have the same "New Mail" notification. I've uploaded a video in case it helps the development team, to show what's going on.
Having the same issue here, it started after recieving fleet mail (which contained an item) and after reading the mail the mail notification remained, even after deleting mail on all characters and checking for any returned items from the exchange. I resolved the issue by mailing myself a random item, the notification was gone for a while after I retrieved it from the mailbox. But after exchanging some duty officers by mail in between my characters the issue has returned and can't get rid of it anymore.
All too often, when we see there's no mail, we'll exit out of the mail screen and not let the system register that we've viewed it.
If you let the mailbox sit on your screen for 5-10 seconds, it should give plenty of time to update the indicator, hopefully letting it disappear.
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Managed to get rid of the notification last night, mailed myself an item and read the mail directly after on the same character. Didn't retrieve the items from the mailbox because I wanted to use them on another character, but at least the notification is gone now.
Comments
https://youtu.be/NwwZ9Qe0B10
If you let the mailbox sit on your screen for 5-10 seconds, it should give plenty of time to update the indicator, hopefully letting it disappear.