What is the truth regarding the conversion of Umar ibn al-Khattab to Islam? Did he embrace Islam due to the Prophet's supplication, or after hearing the Qur'an from his sister or from the Prophet?

There is no authentic, confirmed narration that precisely describes how Umar ibn al-Khattab, may Allah be pleased with him, embraced Islam. The narrations about him hearing the Quran in his sister's house, or hearing the Prophet, peace be upon him, recite near the Kaaba, or the Prophet listening to him and declaring his Islam, all have weak chains of transmission. Despite the weakness of these narrations, most of them agree that the reason for his conversion was his hearing verses from the Holy Quran, which aligns with the conversion of the majority of the Companions during the Meccan period. The supplication of the Prophet, peace be upon him, "O Allah, strengthen Islam with the one of these two men whom You love more: Abu Jahl or Umar ibn al-Khattab," and "O Allah, strengthen Islam with Umar," was indeed answered by Allah facilitating the means that led Umar to Islam, such as opening his ears and heart to the verses of the Quran.