Azure AD Connect needs to write back the msExchMailboxGuid attribute since Exchange 2013 CU12 now requires it to be populated.
After upgrading on prem Exchange Servers 2013 servers to CU12 or 13 you will get an error 500 when trying to open the EAC if the Exchange Admins mailbox was created on Exchange Online and does not have the msExchMailboxGuid attribute populated in AD.
Currently Azure AD Sync does not write back this attribute so customers have to write their own sync scripts. I would recommend either adding this as a write back attribute or fixing the Exchange 2013 EAC not to require this attribute for admins with remote mailboxes
This is related to the mailbox anchoring change
Exchange Management Shell and Mailbox Anchoring
One user cannot connect to ECP on Exchange 2013 server (on-prem)
Jeremy Bradshaw commented
Darn, I'm out of votes. In Exchange 2010/2016 coexistence, I don't see this error you mention, but I do know of this KB article:
The only thing is, MS won't be rushing to fix that, since that issue only prevents offboarding from EXO, ha ha ha..
ExchangeGuid, a.k.a. msExchMailboxGuid, should totally 100% be included in the Hybrid Exchange writeback properties list.