Is “In official histories, Chen Kai, as the grandson of Chen Qiudao, escaped death, while in the novel, as a general under Tian Hu, he was beheaded by Song Jiang” true?

The official history records that his grandfather Chen Qiudao was killed by rebels, and the whole family perished, 'only the grandson Kai escaped by hiding in the mountains and valleys' (History of Song, Biographies of the Loyal, Part 3); the novel, however, writes that as Tian Hu's defender of Qinshui, after defeat, 'the soldiers and civilians captured Yangcheng defender Kou Fu and Qinshui defender Chen Kai, and brought them to the camp', and finally 'they were immediately beheaded as a sacrifice to the flag' (Water Margin, Chapter 94).