The conjecture has been verified already for large enough n that it can be applied without a real proof; a proof won’t give any new “real world” applications. Like with Fermat’s last theorem, the real interest in a proof is in the ideas that would be developed along the way. (For example, on the way to proving FLT Wiles established the modularity theorem for a class of elliptic curves, and his work was extended to a proof of the full modularity theorem in 2001.)