Now, more on this:
Both Dr. Ebubekir Sifil and Professor Hoover relay the following information (which I copied from Hoover's paper as it is already in English):
The Beginnings of Ibn al-Qayyim's Deliberations
Ibn Qayyim al-Jawziyya reveals how he first broached the question of everlasting chastisement with Ibn Taymiyya in an autobiographical note found in his Shifa' al- alil (Healing of the Sick):
I had asked Shaykh al-Islam [about everlasting chastisement]. He said to me, "This issue is very great", and he gave no reply concerning it. Some time had passed after that when I saw in the commentary of 'Abd b. Hamid [or Humayd] al-Kiththi one of those traditions I have mentioned. So, I sent the book to [Ibn Taymiyya] while he was in his last session (fi majlisihi al-akhir). I marked that place [in the book], and I told the messenger, "Say to him, "This place is difficult for him, and he does not know what it is." Then, he wrote his famous work about it. whoever has the grace of knowledge, let him bring it forth, and above each one having knowledge is one who is All-Knowing (pp. 564-65).
It appears that Ibn Taymiyya was not sure how to respond to Ibn al-Qayyim's first inquiry on the duration of the Fire. He only answered that the question was "very great." Ibn al-Qayyim's second inquiry was prompted by reading the commentary of 'Abd b. Hamid al-Kissi (or al-Kiththi as he writes), a ninth-century Hadith scholar from Kiss near Samarqand (d. 249/863)7 A tradition related by 'Abd b. Hamid puzzled Ibn al-Qayyim. So, he marked the spot in the book and sent it to his teacher via messenger. This occurred "while [Ibn Taymiyya] was in his last session," presumably near the end of his life. In reply Ibn Taymiyya composed what Ibn al-Qayyim calls his "famous work."
The "famous work" in question is Al-Radd 'ala man qala bi-fana' al-janna wa al-nar.