How authentic is the popular story about Satan tempting Prophet Ayyub's (peace be upon him) wife, and her attempt to prove her loyalty to him by cutting off her nose and cheek?

This story is not mentioned in the exegesis of the verse: (And take in your hand a bundle [of twigs and strike therewith and do not break your oath]. Indeed, We found him patient, an excellent servant, indeed he was one who oft-returned [to Allah].) [Surah Sad / 44]. Furthermore, it is strange and reprehensible, and it is not characteristic of righteous believing women to do such things to themselves, nor is it something that Allah has permitted for believers. Satan is terrified by a servant’s adherence to the path of his Lord, and his fortification through remembrance of Him, and seeking refuge with Allah from the accursed Satan.