What is the evidence that a fasting person breaking their fast by expelling their saliva or phlegm to the outer lips and then returning and swallowing it, and do all madhhabs and scholars agree on this view?

If saliva separates from the tongue and reaches the lips, swallowing it breaks the fast of the fasting person, because it has become like any other liquid. Scholars have unanimously agreed that whoever expels his saliva from his mouth and then swallows it has broken his fast. As for accumulating saliva in the mouth until it becomes much, and then swallowing it, there are two opinions: that it breaks the fast, and that it does not. The fasting person also breaks his fast if he swallows another person's saliva, or if he swallows his own saliva after it has exited to his clothes, or between his fingers, or between his lips.