Or is Aharon's Staff in Teverya?

The previous post mentions the Rambam's view that the staff of Aharon went along with the Aron.

Sefer Zerubavel, however, gives an alternative viewpoint (text in parenthesis comes from Hashem1.net):

והמטה אשר יתן ה’ לחפצי-בה, אם מנחם, משוקד הוא, והוא נטמן ב-רקת (שם של עיר מקראית), והוא מטה אהרן, ומשה, ודוד מלך ישראל. והוא המטה אשר פרח באוהל מועד, ויציץ ציץ, ויגמול שקדים. ואליהו בן אלעזר (אליהו הנביא שהיה פנחס בן אלעזר הכהן) גנזו ב-רקת, והיא טבריה, ושם גנזו משיח בן אפרים.

And the staff which Hashem will give to Heftzi-va, the mother of Menahem, is almondy, and it is hidden in Reket (the name of a Biblical city) and it is the staff of Aharon and Moshe and of David, king of Israel. And that is the staff which blossomed in the Tent of Meeting, "And it bloomed into a bud and grew almonds". And Eliyahu the son of Elazar (Eliyahu the prophet, who was Pinhas, the son of Elazar Hakohen) hid it in Reket, which is Teverya, and there they hid Mashiah ben Ephraim.

(I'm assuming the fourth-to-last word is Ganezoo - not Genazo, but let me know if you think otherwise, and can explain it.)

(Also, I'm aware that these late Midrashim like Sefer Zerubavel were written in anticipation of the Ge'ula in the 7th century, (or perhaps, the 11th century) and the Ibn Ezra was against relying on it.)