I was not familiar with Listary or ActiveHotkeys before, so your post was a bit confusing until I did some searching.
At first I thought "Listary" was a misspelling and you were talking about Opus listers. After searching for ActiveHotkeys and finding same thread you had posted into about it and Listary, things became clearer...
To avoid confusion and make life easier, if you mention other programs please include a few words explaining what they are and where find information on them. (Unless it's something really well-known like Word or Photoshop, of course.)
Did you try it? You only mention trying Listary with ActiveHotkey but that doesn't imply anything about Opus.
I just tried it and ActiveHotkey has no problem reporting global hotkeys installed by Opus (e.g. the default Win-O hotkey).
Note that Opus disables global hotkeys while the Customize window is open. (That way if you do something silly, like map a global hotkey to the cursor keys or Enter or whatever, then you can still easily clean up the mess once you get into the Customize window.) So if you are looking at ActiveHotkey while also looking at the list of hotkeys in Opus, ActiveHotkey won't detect Opus's hotkeys because they are inactive at that moment.
That's a separate issue...
When a hotkey is "global", all that means is that it will work regardless of which window or application is active.
If you create a global hotkey in Opus then it is definitely global. It's installed using the Windows global hotkey API and if you push the key while another app is active then it works. Can't get much more global than that.
By the nature of the Windows API, global hotkeys require the program that registers them to keep running. (The hotkey event has to be sent to a program or window.)
One of the reasons dopus.exe stays running in the background by default is to allow global hotkeys to work even when you have closed all of the Opus windows.
So, unless you have configured Opus so that it does not run dopus.exe in the background, your global hotkeys should be global and should work at all times.