Did Rabbeinu Yonah write Shaarei Teshuva in response to the burning of the Gemara?

 https://judaism.stackexchange.com/questions/66759/did-rabbeinu-yonah-write-shaarei-teshuva-in-response-to-the-burning-of-the-gemar

It is widely believed that Rabbeinu Yonah played a part in having the Rambam's works burned around 1232, and that after the 1242 burning of twenty-four wagonloads of Gemaras he felt that it was a Divine punishment for the first burning. He wrote Shaarei Teshuva to express his remorse.

This is a citiaion from "The Chosen Path" - a history of Medieval Jewry by Binyamin Sendler: R’ Yonah b. Avraham certainly saw it in this light. Following is an extract from the letter that his student R’ Hillel HaChasid (of Verona) wrote to R’ Yitzchok HaRofeh: Letter by R’ Hillel HaChasid

R’ Yonah of Barcelona (originally of Gerona) was the primary instigator of this controversy that brought about all the calamities that befell the Jews in France. At that time R’ Yonah’s heart melted within him, and he vowed to make a pilgrimage to the Rambam’s grave and remain there for seven days, begging forgiveness from the Rambam for slandering his works. He publicly confessed in these words: “I am embarrassed and I regret that I opened my mouth against our holy master R’ Moshe b. Maimon and his works. I say now for all to hear: Moshe is true and his Torah is true and we are all frauds ....