If a new message is sent in 4.0, the total badge count should reflect your settings and appear on the left-hand side.

It doesn’t make sense to have a disconnect between messages. Right now, I have to hover over Messages just to see if something is new, which then shows a badge count inside Chat/Messages. At the very least, there should be some kind of indication. I would never have known i had received a chat message unless i manually click in.

Inbox gets it right, just not sure why no indication for Chat.