^It is true - BUT - most noses don't have a perfectly straight septum and many have turbines problems etc - stuff that can affect breathing.
In many cases, a singer's voice gets BETTER after a nose job because the breathing improves. But usually, it just stays the same as before the surgery. (For example - great singers like Shirley Bassey, Ofra Haza, Taylor Dayne, Tina Turner, Dolly Parton, Toni Braxton, Kristin Chewnoweth and many others had nose jobs and their voices remained completely the same) I had a nose job 9 months ago and had no change in my voice, either.