Given your teacher’s background, if he recommends submitting a supplement then I think it’s fine, especially if you plan to make music a major part of your academic curriculum.
Here’s a suggestion. Contact Claudio Jaffe and ask if you can send him a sample to see whether he think you should submit a music supplement to Yale. Claudio was the Brazilian student of Aldo Parisot’s who I mentioned earlier who went to Yale and majored in music ( and also did a Masters of Arts and Doctors of Arts in music performance at Yale). I haven’t talked to him in 30+ years, but he was a very nice and approachable guy. Here’s his email:
Here’s his website:
He would probably be aware of your teacher as well, as Parisot and Starker were close friends, and there was a lot of cross-pollination. I shared a house with 2 Starker students when I went to Brazil with Mr. Parisot, and played in masterclasses for Starker when he visited Yale (he had a sardonic sense of humor, and could be brutal).