Did Rabbi Da Modena believe in gilgul?

The Jewish belief in gilgul (literally, reincarnation of the soul) is one fraught with controversy. While it has since become accepted as a basic tenet of Orthodox Judaism, such was not always the case. Among the more famous opponents of this belief is the famed Rabbi Saadiah Gaon, [1] Rabbi Hasdai Crescas and Rabbi Shalom Strashun (Rashash) [2]. R' Saadiah assailed it strongly and claimed it originated from eastern paganism. A lesser famous but equally fierce opponent of gilgul was -one of my favorite Jewish historical figures- Rabbi Leone Da Modena (see my previous posts on him here and here ). However, there is an interesting twist to this. In the book Shem Hagedolim of Rabbi Chayyim Yosef David Azulai (known as the CHIDA) under the entry of Rabbi Yehuda Arye Mimodena, he relates an strange and interesting story. The Chida (pictured) writes (translation mine):

I have seen the latter’s autobiography, 'Chayei Yehuda' (The Life of Judah) in manuscript where he writes that everything he endured in his life was for the best. The Rabbi writes that in his early days he did not believe in Gilgul, then something occurred which made him change his beliefs. His (Da Modena's) neighbor gave birth to a son and within a month the infant took violently ill. When the child reached six months, it was clearly apparent that he at death's door, so the neighbor called him (Da-Modena) to the infant’s bedside to recite Psalms and read from the Torah -as was the custom in Italy at the time. As he was reciting some verses, the child opened his eyes wide and shouted “Shema Yisrael” and his soul left his body. Henceforth, the Rabbi changed his view on gilgul, for his own eyes saw a six month old sickly infant recite the words of the shema like an adult! [3].

This story seems quite strange, not to mention very uncharachteristic of R' Da Modena for obvious reasons. I have read the English translation of Da Modena's fascinating autobiography and I do not recall coming across this story anywhere (though I still have not had the opportunity to read the original Hebrew version). Furthermore, I find it interesting that R' Azulai -the great Kabbalist and proponent of Hassidim- lists him in his Gedolim compendium and refers to him with honorific terms, considering R' Da Modena's radical views regarding Kabballah and Zohar which -as far as we know- he never deviated from.

Notes:

[1]. See his Ha-Emunot V'Deyot 6:3

[2]. See here page 11 for interesting biographical informantion.

[3].שם הגדולים, עג