We started running across this issue since update 15.35 was released. It seems to be happening to random machines right now. Ive tried the basic stuff:
Recreate the email profile
Clicking the sync button in outlook (surprisingly this fixed it for awhile)
Very basic troubleshooting. Users are still complaining. It's not happening to everyone. Was wondering if anyone has seen this behavior. I suppose the next thing is to test the user profile by creating a new one.
I've worked with one person that has syncing stop with Outlook 16.34 and 16.35. Only solution was to remove and add the Exchange (on-premise) account. "Works right up until it doesn't." I haven't tried it with 15.36 yet.
I did try to reduce the message counts in various folders and subfolders, but I am worried the counts are still too high. I had hoped to have enough time to work with them on getting the root mailbox and all subfolders to below 5000 messages per folder and to not have folders more than three deep - I think those numbers are some of the suggestions I've seen.
Also, I haven't enabled logging, which might turn out to be the fastest way to uncover the problem: https://support.microsoft.com/en-us/help/2872257/how-to-enable-logging-in-outlook-for-mac