After the implementation of #273605, it is possible to use "Next Active Tab" as a powerful global shortcut that both puts focus on Konversation and goes to an active tab. The idea is this: if you have Konversation minimized on, for example, #konversation, and a new message arrives on #kde, you could read it by clicking the systray icon then using the Next Active Tab "normal" shortcut. But with the next version it will be possible to just use a Next Active Tab global shortcut to do that.

Therefore, the following should work to follow all new messages even when Konversation is minimized: use Next Active Tab as many times as there are active channels, then press Alt+F4 to set Konversation back aside. Possibly putting markers at some point.

This procedure does work in some cases, but unfortunately not all the time, as I found. The problem is that the current tab is never considered active. This is "normally" desired, but not when Konversation is minimized. The following situation can happen.

You're still on #konversation and #kde, Konversation is still minimized, and a new message arrives again on #kde. The difference is that this time a new message also arrives on #konversation. When you use Next Active Tab again, Konversation gets focus as expected, but also changes to #kde. Changing channel is not necessarily a bug, but the problem is that #konversation is not highlighted (green) as it should. You read the message on #kde, but you have no idea that a new message also arrived on #konversation (unless you saw the notification...).

Although the problem is obvious, the solution is not to me. It is clear that when Konversation is not shown, the tab shouldn't change. But there are cases in between where it's not clear whether it should change, for example if I'm typing in Konqueror and seeing Konversation next to it. It seems this need a redefinition of what an active tab is.

I would have suggested that the current tab is made active in the same cases as new messages on it cause notifications, but I realize that all messages always cause notifications.
This will probably need a conservative solution.