To what extent is the hadith "Whoever has a female [child] and does not bury her alive, nor prefer his male child over her, Allah will admit him into Paradise" authentic?

The hadith was narrated by Abu Dawud, Ahmad, Ibn Abi Shaybah, Al-Bayhaqi, Ibn Abi Dunya, and Al-Hakim. Its wording is: "Whoever has a female child born to him, and he does not bury her alive, nor humiliate her, nor prefer his male child over her—meaning over the female—Allah will admit him to Paradise because of her." Al-Hakim authenticated it, and Al-Dhahabi agreed with him. Dr. Najm Khalaf mentioned that its narrators are trustworthy, except that in its chain of transmission is Ziyad ibn Hudayr, about whom there is some discussion, and he was named by Ibn Abi Shaybah and Al-Hakim.