Does swallowing saliva that has carried phlegm in the throat invalidate the fast, knowing that expelling that phlegm was difficult, and according to the Madhhab of Imam Al-Shafiʽi?

Summary of the Fatwa (definition):

If you are afflicted with waswas (obsessive whispering), you must ignore it and pay no attention to it. You should arm yourself with strong will, sincere determination, and seek Allah's help. Regarding your specific issue, swallowing phlegm does not break the fast unless it reaches the apparent part of the mouth (the articulation point of kha and ha). If it remains within the inner part (the articulation point of hamza and ha), it is not obligatory to expel it, and its return to the stomach does no harm. As long as you are not certain that it reached the apparent part, the default is that your fast is valid, especially since you suffer from waswas. Some scholars have favored the view that swallowing phlegm does not break the fast even if it reaches the mouth, and this is a view held by some Shafi'is.