Stage Retriever is a way for you to share multiple screens at once to any meeting app, this way you can focus on what you are presenting and you will not have to bring each window to the screen you initially choose to share.
Apparently if you don't have things like Microsoft Powertoys Awake feature enabled the application will for some reason stop rendering on windows as soon as you to a different virtual desktop screen.
Possible fixes: read the PowerToys source code to implement the same Syscall they use for Awake or play with the powerSaveBlocker API.
The screen recording related APIs seem to just not be available there, it has to be still investigated, probably electron needs to be updated to fix the issue.