Is “Whether Xiahou Ba fought with Wei forces before surrendering to Shu Han” true?

In official history, Xiahou Ba, because Cao Shuang was executed, fled directly to Shu in fear: "He had been favored by Cao Shuang. Hearing of Shuang's execution, he became suspicious and fled into Shu" (Records of the Three Kingdoms, Book of Wei, Biographies of the Xiahou and Cao families). The novel adds a plot where he raises troops in rebellion, describing that only after being defeated by the pincer attack of Guo Huai and Chen Tai did he flee to Shu Han: "Ba was greatly defeated and fled, losing more than half of his troops; finding no plan, he then went to Hanzhong to surrender to the Later Lord" (Romance of the Three Kingdoms, Chapter 107).