Detect Private Window or listen for its Hot Key

Are there any system properties associated with Private Windows? For instance, I was hoping something like MapName_Active would report if the map associated with a window is open, but I tried every imaginable name combination and there appears to be no such property. I’m basically trying to trigger actions whenever a private window is opened/closed or its hotkey is pressed. I guess another approach would be to have a piece always listen for the hotkey, but it doesn’t seem like there is a way to do that either. Has anyone figured out a creative way to achieve this?

The only thing I can think of is to create a fake private window button that is actually a GKC that triggers the hotkey for the real private window button plus whatever other actions I want. The undesirable side effect would be that two instances of each button would appear on the toolbar: the fake private window button and the real private window button (which could be reduced to a nub with no icon or text but cannot be hidden).