Did the Christians of Najran embrace Islam through correspondence?

When the verses of Mubahala were revealed, the Prophet, peace and blessings be upon him, invited the Christians of Najran to it. They consulted among themselves and said: "If he debates us with his people, he is not a prophet. But if he debates us specifically with his household, then we shall not debate him, for he is truthful." So when morning came and the Prophet, peace and blessings be upon him, came out with his household, they realized he was a prophet and said: "If he debates us, Allah will surely destroy us." So they came forward and said: "We will give you satisfaction, so exempt us from the Mubahala." And about twenty men among them embraced Islam later on.