Is “The reason and manner of Crown Prince Dan of Yan's escape from Qin” true?

The Records of the Grand Historian only records that the King of Qin treated Crown Prince Dan of Yan poorly, and Dan fled back due to resentment ("The King of Qin treated Crown Prince Dan of Yan poorly, so Dan resented and fled back" - from the Biography of Assassins). However, the Chronicles of the Eastern Zhou Kingdoms adds a series of fictional plots: Crown Prince Dan taught the King of Yan to feign illness and summon him back; the King of Qin deliberately made things difficult by saying, "If you want the Crown Prince to return, unless the heads of crows turn white and horses grow horns!" Then Crown Prince Dan shouted loudly, causing "all crow heads to turn white," but the King of Qin still would not let him go. Finally, Crown Prince Dan "changed his clothes, disfigured his face, became a servant, and sneaked out of Hangu Pass" (from Chapter 106).