Existence of Dickson polynomials among the associated polynomials

Let $p$ be a prime, let $n+1$ be a prime with $n+1\neq p$, and let $(n,k)=1$ and $(n+1,p^2-1)=1$. Set

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

For $a\in\mathbb{F}_{p^{nk}}$, define $a_i=a^{p^{ik}}$ for $0\leq i\leq n-1$ and

$$h_a(x)=x\prod_{i=0}^{n-1}(x+a_i).$$

Existence conjecture. There exists $a\in\mathbb{F}_{p^{nk}}^*$ such that $h_a(x)$ is a Dickson polynomial of degree $n+1$ over $\mathbb{F}_{p^k}$. This conjecture was subsequently proven in odd characteristic, so it is recorded as solved.