I agree that is an issue when group inboxes are heavily used.
However, a better solution might be to surface the groups themselves as they are nicely managed from the group page:
Have you considered a GROUPS section (default turned off) in the sidebar which lists groups who the user is a member of? And then navigates to the Group message tab by default (with a blue indicator if there is a new message)? I know that they can navigate to https://meta.discourse.org/g?type=my, but that adds at least one more step and doesn’t surface the messages.
Personally, I’d find this very useful for managing groups and their messages more effectively, and enabling users to find their groups more easily.