Complete permutation polynomials from exceptional polynomials

We classify complete permutation monomials of degree qn−1q−1+1 over the finite field with qn elements in odd characteristic, for n+1 a prime and (n+1)4<q. As a corollary, a conjecture by Wu, Li, Helleseth, and Zhang is proven in odd characteristic. When n+1 is a power of the characteristic we provide some new examples. Indecomposable exceptional polynomials of degree 8 and 9 are also classified.