Is “Hit by an arrow at Zengtou City” true?

In the novel, Chao Gai dies after being struck by an arrow in the face while attacking Zengtou City. In the three passages of the "History of Song" that record Song Jiang's band (the Annals of Emperor Huizong, the Biography of Zhang Shuye, and the Biography of Hou Meng), none of the thirty-six are named, and both Chao Gai's existence and his death in battle lack any basis in official history, being a fictional creation of the novel. The original text reads: "不期一箭，正中晁盖脸上，倒撞下马来。" ("Water Margin", Chapter 60)