Is the story of Abu Jahl killing a Muslim woman during the Makkan period authentic, or does it have no basis?

It is well-known among biographers that Abu Jahl was the killer of Sumayyah bint Khabbat, the mother of Ammar ibn Yasir, and that she was the first martyr in Islam, despite the reports regarding her killing being mursal (lacking a continuous chain of narration).

Ibn Sa'd narrated in "Al-Tabaqat" from Mujahid that the first seven people to openly declare Islam included Sumayyah, and that Abu Jahl stabbed her and killed her, and that she was the first martyr in Islam. Ibn Kathir and Ibn Hajar mentioned that this narration is mursal, but it is famous and accepted in historical accounts and military expeditions, especially since it does not contain anything objectionable. Sumayyah was a slave of Banu Makhzum, and Abu Jahl was among the most severe tormentors of the Muslims from that tribe.