We have an Exchange 2013 / Hybrid O365 deployment with Office 2016 Enterprise E3 ProPlus installed on Server 2012 R2 RDS Servers.
We have about 10-13 users per RDS Session Host.
We are running Outlook in cached mode via Group Policy, because we were experiencing lag and outbox sending issues with online mode.
A few users started seeing delays with received email in Outlook. They would receive it on their iPhone or in OWA, then in Outlook 30 minutes later.
The Outlook .OST file has been fully synced for at least a week. There are no new .OST files being created.
If the user closes Outlook, then re-opens it - they get a few recent emails but Outlook goes back to delaying them. Outlook will say "all folders up to date" at the bottom, even though they are not.