This can be tested in app or webapp and should be tested by a dev. Login into LMS and go to Site administration > Mobile features > Disabled features.  Then, disable messaging and notifications. Login into the web app or with a device connected to your computer and check network requests. Wait to the execution of the CRON (it takes 1 min in webapp and 10 in mobile) and when the execution is done, the app shouldn't have called the refresh notification ( core_message_get_messages ) or messaging WS ( core_message_get_unread_conversation_counts, core_message_get_received_contact_requests_count )