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I would verify that their Exchange account is the only account within outlook. From a few different places it looks like it could be something stuck in Drafts too (although you said you checked there). Possibly removing the account and re-adding it in Outlook could be the quick fix to it (although they would lose their saved autocompletion which a lot of people complain when they lose it).
Jan 9, 2019 - Exchange 2007 mailbox is not supported by Outlook 2016/2019 There is. Downgrade your Office 2016/2019 back to Office 2013 following the steps below. If the record cannot be obtained from local domain, the SCP lookup fails. SecuriSync®: Updating Desktop Clients SecuriSync®: Clients Cannot.
Im running 2016 on probably 25 Mac's and haven't seen this issue yet, but I have had weird issues with Outlook in general that aren't worth the time messing with and just adding the account back again will fix saving all of our times.