Though not because of Mohammedan religious sensitivities. After all, the Third Temple might descend upon us from on high at any moment, without prior warning. How will you run without shoes on? Don’t get crushed! (Besides, what would we do with the Tumah?)
It is said Rabbi Aaron of Belz zatzal avoided entering Har Hazeisim lest this mountain suddenly split open in the middle, as per Zecharya Hanavi, so there’s a simple kal vachomer (assuming the story true)!
Find the Fallacy…