I'm not so sure about that. I knew a Garrett in high school who was never Gary. "And by the way, dearie, your punctuation sucks canal water!" -The Ghost of VivianVance
At my prior workplace, I had a co-worker whose son was named Garrett and she always referred to him as Garrett. I don't know for a fact that he was never called Garry by anyone, but at least she never referred to him that way.