Mail app updating offline disk cache
If quitting and relaunching Mail isn’t reasonable, Apple has offered their own workaround to deal with the new mail issue until a future update comes out to fix the problem: taking the troubled account offline, then bringing it back online, thus forcibly reestablishing a connection between Mail app and the remote email server.Here’s how to do that: This forces a connection to be made between the Mac Mail app and the remote mail server, thereby downloading new emails and updating the inbox as expected.Next, when a user attempts to connect to Exchange to retrieve mailbox data, the device key is again passed from the device to the Outlook service over a TLS-secured connection, where it is used to decrypt the password in runtime compute memory.Once decrypted, the password is never stored in the service or written to a local storage disk, and the device key is once again wiped from memory.This appears to be a connection issue limited to how the Mac Mail app interacts with specific email providers, which is why it’s not impacting every user.
The device key, meanwhile, is wiped from memory and never stored in the Outlook service (the key is only stored on the user's device).
After three days of inactivity, the Outlook service will flush a decrypted password from memory.
With the decrypted password flushed, the Outlook service is unable to access a user's mailbox.
Only users who have been effected by new mail not showing up reliably should need to bother with this, and if you haven’t noticed any issues so far then you’re probably not impacted.
Before going any further, you should double-check to make sure you have installed any software updates and mail updates that are waiting, those updates may very well fix the issue you are experiencing with new emails not showing up.