Is it true that Umar ibn al-Khattab exiled Nasr ibn al-Hajjaj due to his excessive beauty, and what was the reason for that?

The summary of the story is that Umar ibn al-Khattab, may Allah be pleased with him, was patrolling Medina at night when he heard a woman singing verses that mentioned "Nasr ibn Hajjaj." He summoned Nasr and found him to be a handsome young man. So, he had his head shaved, which only made him more beautiful. Fearing that women would be tempted by him, Umar then exiled him to Basra. The reason for exiling Nasr ibn Hajjaj stems from prioritizing public interest over private interest, as inflicting harm on private interest for the sake of public interest is imperative. Furthermore, his exile weakens the cause of temptation in himself and in others, teaches people to combat desires and condemn indecency, and the sanctity of Medina necessitates expelling from it anyone by whom women are tempted.