The supplement is valued for what it shows you can contribute to the mix on campus. But you aren’t under any obligations. The head of the jazz program may look for you though, who knows. Has anyone had experience with this, with, for instance, a faculty member e-mailing a student who submitted a supplement? This person didn’t meet with you and you didn’t ask for any special attention. There is no personal connections. I think you will be fine. And you at least represent POTENTIAL contributions on campus.