My son is 16 and wants to go to a music conservatory. Which of the big ones do you recommend?

He's 16 (17 soon) and a junior in high school and has been playing piano since he was four (although at one point he did stop for a few years). Right now he's learned Mozart's 8th and 10th piano sonatas, he's working on two easier Chopin Etudes, Schubert's Impromptu in E flat major, a few short Bach preludes and the Invention in A minor. He's also completed 11 Etudes from Czerny's Art of Finger Dexterity and three from the School of Velocity. Though, he hasn't performed publicly much, and hasn't participated in any competitions. Are there any chances of certain schools like Juilliard or New England Conservatory if he practices 5-8 hours a day like he currently does?