Update: The "New Release" info is stored in Innovator\Server\temp\VersionServer.xml
This is surprisingly not helpful at all. At first, I thought this might be exactly what I was looking for, but Aras probably relies on some internal functionality to populate this information in Innovator.
That said, I may be overthinking the problem. I can use an init event to display the correct button state on startup. The only thing left to solve is how to update appearance of the button whenever a user starts or stops their timer.
The main challenge seems to be figuring out how to access or interact with a CUI element in the main window header from another context (grid, form, etc.). Unfortunately, this approach does not work because "control" will be undefined.
const key = "com.aras.innovator.cui_default.mwh_header_notifications_button";
const data = control.data.get(key);
