We've noticed that after migrating a mailbox from 2007 to 2013, any search folders in the user's Outlook 2010 client still work but they no longer sync the read/unread status of mail items.  If we read an unread item, the search folder continues to display it as unread.  The workaround is to delete and re-create the search folder.  Any ideas?

Please see the Melon Chen  Reply on this thread it might be helpful

https://social.technet.microsoft.com/Forums/en-US/adf324a4-5764-4509-8e6b-06a077dc2a33/outlook-2013-not-syncing-inbox?forum=officeitpro

Hi,
 
Basis on your post, I notice that cannot sync read/unread status in search folder when we read an unread item.
If I misunderstand your concern, please do not hesitate to let me know.

I want to double confirm whether all user or some special account experience this question.
How about repair Outlook profile or recreate a profile for testing. Meanwhile try to run Outlook to online mode.

Besides, please upgrade Outlook for latest version. If the issue persists, collect the synchronization log in Outlook for further troubleshooting.

Thanks