Is “Did King Huan of Zhou strip the Earl of Zheng of his office, and after being shot and injured while attacking Zheng, entrust his son to Duke Heijian of Zhou on his deathbed?” true?

The 'Records of the Grand Historian, Basic Annals of Zhou (part 2)' records King Huan's clash with Zheng: 'In the thirteenth year he attacked Zheng, and Zheng wounded King Huan with an arrow; King Huan withdrew and returned.' The seizure of the Earl of Zheng's office appears in the 'Zuo Tradition' under the fifth year of Duke Huan ('The King seized the Earl of Zheng's office, and the Earl of Zheng did not attend court'). But the deathbed entrustment to Duke Hei Jian of Zhou is not in the histories: the 'Basic Annals of Zhou' says only that 'in the twenty-third year King Huan died and his son King Zhuang, named Tuo, succeeded him. In King Zhuang's fourth year, Duke Hei Jian of Zhou sought to kill King Zhuang and enthrone Prince Ke' -- a revolt after King Huan's death. The 'Zuo Tradition' separately notes that Ziyi (Prince Ke) had been favoured by King Huan. 'Chronicles of the Eastern Zhou Kingdoms' works these threads into a full deathbed scene, which is the novel's own addition.