Conjecture on Dickson-polynomial auxiliary functions for CPP exponents

Let $p$ be a prime, let $r+1$ be a prime with $r+1\neq p$, assume $\gcd(r,k)=1$ and $\gcd(r+1,p^2-1)=1$, and set

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

The functions $h_a(x)$ are associated to elements $a\in\mathbb{F}_{p^{rk}}^*$.

Dickson-polynomial conjecture. There exist $a\in\mathbb{F}_{p^{rk}}^*$ such that $h_a(x)$ are Dickson polynomials of degree $r+1$ over $\mathbb{F}_{p^k}$.

This conjecture concerns a family of exponents used to construct complete permutation polynomials. It has been proved for several parameter choices, including $p=2$ with $r=4,6,10$, $p=3$ with $r=4,6$, and $p=5$ with $r=6$; it was also verified computationally for $p=3$, $r=10$, and $1\leq k\leq 3$.