I'm seeing a few users using O365 that don't have updating inboxes.
If you check their web mail, their mail arrives in a timely fashion but the inbox it doesn't.
It happens on Outlook 2010 and 2013.
It goes away for a "short time" if you go into
File -> Account -> settings
select the account and choose "Repair" then next next next and restart outlook.
One thing in common with the users so far affected is that they all have multiple accounts listed in account settings (Share mailboxes and PA's looking after their Executive mailboxes)
I have also tried Profile deletion and rebuilding and I have tried turning off "Cached mode" restarting the client, then turning cached mode back on again.
Any idea?
