I suppose encouraging each member to create their own private custom section and populate it with their inboxes may get you slightly closer.
But if this is not going to be picked up as a core feature what would be the best way to build a customisation to achieve it? Would it need to be a plugin, or a maybe even just a theme component? Or is it just not possible?
I don’t want to raise too much hope here, but I think adding a sidebar section with the group inboxes including unread indicators is possible with a Theme component. At least I got this: