First thing is we need to make sure we understand that the primary SMTP address and username, first name, last name are user attributes in AD. If you are running a hybrid environment with Azure AD Connect, then all changes need to be made on-premises and then synced to Office 365/Azure AD.
Since you are syncing all accounts from on-premises AD to Azure AD, in a Hybrid environment, your active directory is synced with Office 365 using Azure Active Directory Sync tool (DirSync). Therefore, your on-premises active directory is the authority for making any changes to any of the user attributes, and you need to manage the attributes of the user accounts from your on-premises active directory.
Proxy address first gets synced to Azure AD and post that this value gets back synced to EAC. Proxy address is an attribute which is used in EAC.
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