Is “Han Dang rescued Huang Gai from the water during the Battle of Red Cliffs” true?

In Chapter 50 of the novel 'Romance of the Three Kingdoms', it is written that Huang Gai was hit by an arrow and fell into the water during the Battle of Red Cliffs, crying out 'Yigong, save me!' Han Dang recognized his voice and rescued him, and 'took off his own battle robe and put it on Huang Gai.' The main text of 'Records of the Three Kingdoms' only records that Huang Gai 'proposed the plan of fire attack,' with no mention of falling into the water and being rescued. The episode of Han Dang rescuing Huang Gai actually comes from the 'Book of Wu' cited in Pei Songzhi's annotations, and the historical detail is that Huang Gai was first rescued by someone else, and only then called out to Han Dang.