Is it true that Abu Lahab supported the Prophet (peace and blessings of Allah be upon him) after the death of his uncle Abu Talib and before his departure to Ta'if?

The story of Abu Lahab’s support for the Prophet (peace be upon him) after the death of Abu Talib is problematic and has not been authenticated by anyone. Its summary is that after the death of Abu Talib and Khadijah, the Quraysh persecuted the Prophet (peace be upon him). Abu Lahab came to him and said: "O Muhammad, go forth with what you intend, and whatever you were doing when Abu Talib was alive, do it now. No, by al-Lat, no one will reach you until I die." When Ibn al-Ghaitalah reviled the Messenger of Allah, Abu Lahab defended him, which made the Quraysh praise him for upholding kinship ties. So, the Quraysh stopped harming the Prophet (peace be upon him) for a few days. Then, Uqbah ibn Abi Mu'ayt and Abu Jahl came to Abu Lahab and asked him about the place where his father entered (the afterlife). The Prophet (peace be upon him) answered: "With his people." When they informed him that he (the Prophet) claimed his father was in hell, Abu Lahab asked the Prophet (peace be upon him) if Abd al-Muttalib would enter hell. The Prophet replied: "And whoever died upon what Abd al-Muttalib died upon, entered hell." At that, Abu Lahab said: "By Allah, I shall never cease to be your enemy forever," and his enmity and the enmity of Quraysh towards the Prophet (peace be upon him) intensified.