This is something I have been craving for for a long time, I described it here as #4 with the exact same use case. Offering "private support" as part of a public support community
I believe group inboxes are inherently flawed in a number of ways (like this)
It should be doable with a plugin although it might take some time to get right. I just have not found the time or justification for the effort yet (if you have a budget and/or development skills, let’s talk)