Perhaps you should remind your friend of Samson who said to Delilah, “A razor has never come upon my head; for I have been a Nazirite to God from my mother’s womb. If I be shaved, then my strength will leave me, and I shall become weak, and be like any other man.”
(Judges 16:17)
Concerning Nazirites, the Bible says, "And the LORD said to Moses, "Say to the people of Israel, When either a man or a woman makes a special vow, the vow of a Nazirite, to separate himself to the LORD, he shall separate himself from wine and strong drink; he shall drink no vinegar made from wine or strong drink, and shall not drink any juice of grapes or eat grapes, fresh or dried. All the days of his separation he shall eat nothing that is produced by the grapevine, not even the seeds or the skins. “All the days of his vow of separation no razor shall come upon his head; until the time is completed for which he separates himself to the LORD, he shall be holy; he shall let the locks of hair of his head grow long.” (Numbers 6:1-5)
Since Jesus was singularly separated to the Lord, it seems consistant with Sacred Scripture that he would have let the locks of hair of his head grow long.