To receive calls and messages via Swyx Mobile for iOS it is not necessary to enable background App Refreshing in the iOS settings. It is also not necessary to keep Swyx Mobile for iOS active in the background for reachability. The Push Notification Service alone ensures that calls and messages are signaled.

If you don’t want to receive the signaling, you have to sign out from Swyx Mobile for iOS (see also Sign out from Swyx Mobile for iOS) or use the “Do Not Disturb” function of the iPhone. This mutes calls, alerts and messages that you otherwise receive while your device is locked. See also “Do not disturb” under https://support.apple.com/en-us/HT204321