Is the story of Omar Ibn Al-Khattab (may Allah be pleased with him) burying his daughter alive true?

The story of Umar ibn al-Khattab (may Allah be pleased with him) burying his daughter alive in the pre-Islamic era (Jahiliyyah) is not authentic for the following reasons:

1. The story is not mentioned in any books of Sunnah (definition), Hadith (definition), or history. It is an fabrication of the Rafidah (Shi'a). 2. His daughter Hafsa was born five years before the Prophetic mission, and she was not buried alive, which indicates that burying daughters alive was not his custom. 3. Umar narrated the Hadith about the expiation for burying daughters alive, attributing the act to Qays ibn Asim, without mentioning anything about himself. 4. Even if the story were hypothetically true, Islam nullifies what came before it. If Shirk (definition) (polytheism) is forgiven, then burying daughters alive is even more deserving of forgiveness. 5. Al-Aqqad doubted the story due to the uncommonness of burying daughters alive among Bani Adi and the lack of anyone mentioning details about this alleged daughter.