Does swallowing saliva after it has reached the outer lip invalidate the fast, and do the madhhabs agree on this, and is there a permissive opinion that can be acted upon for ease?

If a fasting person intentionally swallows their saliva after it has reached the lips, they break their fast according to the Shafi'i and Hanbali schools of thought, because it has separated from its original place and thus become like other detached substances. The Hanafis, however, are of the view that it does not break the fast unless the saliva has completely separated from the mouth and then been re-swallowed. But if it remains connected, it does not break the fast. It is permissible to adopt the view of the Hanafis for someone who suffers from obsessive doubts, although it is more prudent not to intentionally do so, without being overly strict.