Was the convergence upon the Mahdi in the year 1240 AH, as mentioned in the narration, an independent judgment (ijtihad) by Ibn Hajar al-Haytami, or was it a confirmed report? And did other scholars differ with him on this matter by mentioning other years that have not yet arrived?

It is part of the belief of Ahl al-Sunnah (definition) that the emergence of the Mahdi is one of the signs of the Hour. He will rule for seven years, filling the earth with justice. Jesus will descend, kill the Dajjal, and pray behind the Mahdi. Everything mentioned about specifying the time of his emergence is incorrect, because it pertains to the unseen, and no specific timing has been authentically reported from the Prophet, peace and blessings be upon him. Moreover, some of the narrations mentioned in this regard are inauthentic, such as what was stated by Ibn Hajar al-Haytami and Nu'aym ibn Hammad in "Al-Fitan"; this book is weak and cannot be used as proof.