Complete permutation classification for reversed Dickson polynomials of the first kind

Let $p$ be an odd prime, let $1\leq n\leq p^2-1$, and let $D_n(1,x)$ be the reversed Dickson polynomial of the first kind. A polynomial $f$ is a complete permutation polynomial (CPP) when both $f(x)$ and $f(x)+x$ permute $\mathbb{F}_p$. Reversed Dickson complete permutation conjecture. $D_n(1,x)$ is a CPP on $\mathbb{F}_p$ if and only if

$$n = \begin{cases} 2,2p,3,3p,p+1,p+2,2p+1 & \text{if } p\equiv 1 \pmod{12},\\ 2,2p,3,3p & \text{if } p\equiv 3 \pmod{4}\text{ or }p\equiv 5 \pmod{12}. \end{cases}$$

The conjecture is presented as strongly supported by computer searches and follows from the preceding permutation conjecture together with the results in the subsection; its general necessity remains unresolved.