Is the story of Badhan, the Persian governor of Yemen, embracing Islam true?

Ibn Hisham, Al-Tabari, and others mentioned that the Prophet (peace and blessings be upon him) sent Abdullah ibn Hudhafah Al-Sahmi with a letter to Chosroes, inviting him to Islam. Chosroes tore the letter, so the Prophet (peace and blessings be upon him) supplicated that his kingdom be torn apart. Chosroes then wrote to Badhan, his governor in Yemen, to send two men to bring him news of the Prophet (peace and blessings be upon him). Badhan sent two men, who arrived in Medina and presented Badhan’s letter to the Prophet (peace and blessings be upon him). He smiled and invited them to Islam, then told them to return the next day. The next day, he informed them that his Lord had killed their lord, Chosroes, that very night, and that his son, Shiruyih, had killed him. The two men returned to Badhan with this news, whereupon he and his sons in Yemen embraced Islam. Al-Albani authenticated this hadith.