Get-Mailbox does not reflect the correct Server name for Mailboxes on databases for a DAG

We got notified from customers, that the powershell commandlet get-mailbox in Exchange 2010 (see and in Exchange 2013 (see do not always return the correct ServerName  or ServerLegacyDn for the given mailbox. Those customers often rely for their backups or customized mailbox processes on the information, on which server a given mailbox is currently located….