Is “How did King Zheng of Qin obstruct Crown Prince Dan of Yan from returning to Yan?” true?

The Records of the Grand Historian, Biographies of Assassins, directly records Dan's flight: "The King of Qin treated Crown Prince Dan of Yan poorly, so Dan resented and fled back." There is no supernatural element at all. In Chronicles of the Eastern Zhou Kingdoms, Chapter 106, to dramatize the tyranny of the King of Qin and the will of Heaven, it fabricates the difficult conditions: "To let the Prince return, unless crows' heads turn white and horses grow horns, then it may be done!" Ultimately, Dan's resentment soars, triggering strange omens.