The badge count on the iOS app won't refresh
There can be a number of reasons why the badge count won't refresh.
Check iOS Settings > Notifications > BusyCal. Make sure you have given the app permission to update Badges
If you are force-closing the app (i.e. by swiping up from the bottom of the screen, then closing the apps by swiping on them), iOS takes that as a hint that you don't want that app to sync in the background and will not allow any background processing. This will also prevent it from updating badge counts and performing other timely sync related activities. When this happens, you much physically launch the app manually for it to later sync in the background.
You are restarting your phone frequently, or turn it off at night. After a restart, iOS does not auto-refresh any app. You must manually launch the app once for it to be included in the next timely refresh. People have written a lot of articles on not to force-quit or restart the phone unnecessarily: https://tidbits.com/2020/05/21/why-you-shouldnt-make-a-habit-of-force-quitting-ios-apps-or-restarting-ios-devices/
The app may simply be throttled by iOS as it tries to save on battery. Sadly there's nothing we can do about iOS throttling. iOS learns how you use the app, how frequently you force-close it and takes these into account when considering it for a background refresh.
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