John Hyrcanus the Last King and High Priest of the Judahites (The Hebrew tribe of Judah/Isreal) allowed Edomites to live in The Kingdom of Judah if they followed the laws of the Kingdom.
It was not a conversion, it was an acquiescence to the law. There is a HUGE difference. They (The Jews/Edomites) did not become party to the covenant with Yahweh.
1) Because Yahweh never made a covenant with them (Esau and Edom)
2) You can't amend a covenant contract.
3) Yahweh hated the seed line of Esau forever.
Esau is Edom. (Genesis 36:8)
Or as scripture says elsewhere: I loved Jacob but hated Esau. What should we say, then? That God is unjust? Out of the question!
(Romans 9:13-14)
Next, these Edomites killed ALL the Nobles and Priests of the Judahites.
So the last tribe of Israel (Judah) became a remnant in their homeland and there were only small pockets left. Most of them went north into Samaria and Galilee.
That is why Jesus was scared to goto Jerusalem "for fear that the Jews (Edomites) would kill him"
(John 7:13, 19:38, 20:19)