Does swallowing phlegm invalidate the prayer according to those who hold that swallowing it does not invalidate the fast, such as Sheikh Ibn Uthaymeen, may Allah have mercy on him?

Al-Hattab mentioned in "Mawahib al-Jalil" that Al-Lakhmi was asked about clearing the throat (tanaffuh) during prayer, and he replied: "Everything that descends from phlegm into the throat and is swallowed by the accountable person does not invalidate a fast or a prayer—even if he could have expelled it—as long as it does not reach the fauces. However, if it comes out to his mouth and he swallows it, there is a difference of opinion as to whether he must repeat his fast and prayer, like food, or not, since it is not in the same category as food."