Is “After being exiled, Qu Yuan drowned himself by holding a stone, and the people rowed boats to save him and threw zongzi to feed the fish.” true?

The Records of the Grand Historian, Biography of Qu Yuan and Jia Yi, clearly records that Qu Yuan 'held a stone and threw himself into the Miluo River to die'; however, the Dragon Boat Festival customs of dragon boat racing and making zongzi were only later associated with Qu Yuan in post-Southern Dynasties texts such as the Continued Records of Qi Harmony, and novels often anachronistically move these customs earlier.