Process status notifications when the Run tool window is invisible

It would be cool to see the process status notifications when the Run tool window is not visible on the screen (this is a common situation, say, in Rider).

In those cases, the plugin may show a notification that suggests the user to open the tool window if they need to see the stdout (but mind the message length, it should be very succinct).

I'm not sure if it's a good idea to use the IntelliJ standard notification system for that (those notifications may be a bit annoying if you often start processes); it would be interesting to investigate a process of showing a tool window notification without an actual tool window (so, the notification should look similar to the current one, but also be able to pop-up and disappear if the tool window button is hidden from the screen).