It may take some time for changes to automatically sync to your iPhone contacts (up to 8 hours.
This said - here is how to force the app to sync:
1. Go to your iPhone 'Settings'
2. Tap 'Mail, Contacts, Calendars'
3. Tap 'Haystack'
4. Toggle 'Account' from 'On' to 'Off'
5. Toggle 'Account' back to 'On'
All changes should update in your phone contacts.