What is the authenticity of attributing the circulating image of the cloak, turban, and staff to the Messenger of Allah, may Allah bless him and grant him peace, and what is the meaning of the word "burdah"?

Historians have differed concerning the Prophetic burda (mantle) that was in the possession of the Caliphs. Some are of the view that it was the burda of Ka'b ibn Zuhayr, which Mu'awiya purchased. Others hold that it was the burda of the people of Aylah, which Abu al-Abbas al-Saffah purchased. Mu'awiya's burda was lost when the Umayyad state fell. This burda was inherited by the caliphs, who would place it on their shoulders during processions. Some historians mentioned that it is present in Topkapi Palace in Turkey, where Sultan Ahmed I placed it over the throne for blessings, and Sultan Mahmud II left the private chamber, leaving it for the safekeeping of the sacred trusts.