No, checking each mail in inbox, marking all as read and emptying inbox does not help.
This is likely because of the account merge, where my long inactive xbox account had unread messages in inbox. Those messages always show as unopened messages, even though mailbox would be empty. Still the inbox opens after every mission, every transition to hub/relay/orbiter and alt+tabout and back into game.
By far the most annoying bug in this game so far. Not game breaking, but imagine getting stopped, swing to menu, open inbox and hear Ordis complain about unread mail ~50 times per day.