We have the same issue. Exchange 2016 servers, migrated from Kaspersky to EAV 6.3 (about 300 computers). And on 3 of them new e-mails stopped coming (Updating Inbox).
1. On 1 PC: Outlook 2010 x86 installed (Windows 7 x64). I deleted Outlook profile, and re-download mail from server. New messages appeared. About 3 hours later, same computer, new messages stopped coming again.
After installing Outlook 2013 x64 and updating, mail started working normally.
2. On two other PCs (Outlook 2013 x86, Windows 7 x64/Windows 10 x64) also stopped coming mail. Restart of computer didn't help. After disabling Outlook ESET add-in, new messages came.
Yes, 3 PCs from 300 - is not a big percent, but unfortunately it was computers of top management (big mailboxes 10-15 GB) and we temporary disabled ESET Outlook add-in for all.