A characteristic-prime criterion for complete permutation monomials

Let $p$ be an odd prime, let $n+1=p$, and set

$$d=\frac{p^{nk}-1}{p^k-1}+1.$$

Let $a\in\mathbb{F}_{p^{nk}}^*$ satisfy $a^{p^k-1}=-1$. Complete permutation monomial conjecture. The monomial $a^{-1}x^d$ is a complete permutation polynomial over $\mathbb{F}_{p^{nk}}$. The source reports that this conjecture was proven in odd characteristic, so it is recorded as solved.