Is it true that when a pigeon in Medina approaches its time of death, it flies to Mecca and passes over the Kaaba as a farewell before it dies?

The pigeons of Makkah and Madinah have no special characteristic except that it is forbidden to hunt or disturb them within their boundaries, whether for a pilgrim in ihram or not, due to the Almighty's saying: "O you who have believed, do not kill game while you are in ihram," and due to the Prophet's (peace be upon him) saying: "Indeed, Allah made Makkah sacred, and it was not made lawful for anyone before me, nor will it be made lawful for anyone after me. It was made lawful for me only for an hour of a day. Its fresh grass is not to be cut, its trees are not to be felled, and its game is not to be disturbed." And his (peace be upon him) saying: "Indeed, Ibrahim made Makkah sacred, and I made Madinah sacred, what is between its two lava fields. Its thorny trees are not to be cut, and its game is not to be hunted." The claim that the pigeons of Madinah fly to Makkah when their death approaches to bid farewell to the Kaaba is a lie and ignorance, for lifespans are in the hand of Allah, "and no soul perceives in what land it will die."