Nowadays the inbox system is pretty great, but has one bit missing: I can connect to a roleplay or PM on my phone, and forget I have it open at all. The system will consider it connected for hours or days, and I'll receive no notifications about posts to it.
Somehow, and I'm not totally sure how, I'd like to still receive messages to channels where I'm logged in elsewhere on another device, but do not check or have the opportunity to check those messages.
I do know there's something we need to avoid: a "gotcha" time period. The old notification system, for example, wouldn't notify you of anything that occurred within 15 minutes of you exiting the site - doesn't seem like a problem on paper, since 15 minutes isn't very long, but in practice there was enough activity on site that people were losing notifications about PMs and roleplays to that 15 minute black hole.