The ring tone is not contained in the raw contact. When setting an ring tone in the Contacts app, CUSTOM_RINGTONE of the RawContact is not set (it’s only set for the aggregated contact). See RawContacts docs:A custom ringtone associated with a raw contact. Typically this is the URI returned by an activity launched with the ACTION_RINGTONE_PICKER intent. To have an effect on the corresponding value of the aggregate contact, this field should be set at the time the raw contact is inserted. To set a custom ringtone on a contact, use the field Contacts.CUSTOM_RINGTONE instead.
So I guess DAVdroid should not synchronize ring tones, as it only aims to synchronize raw contacts.