Reversed Dickson first-kind permutation polynomial classification

Let $p>3$ be a prime, and let $n$ satisfy $1\leq nleq p^2-1$. Write $D_n(1,x)$ for the reversed Dickson polynomial of the first kind. A polynomial that permutes $\mathbb{F}_p$ is called a permutation polynomial (PP). Reversed Dickson permutation conjecture. $D_n(1,x)$ is a PP on $\mathbb{F}_p$ if and only if

$$n = \begin{cases} 2,2p,3,3p,p+1,p+2,2p+1 & \text{if } p\equiv 1 \pmod{12},\\ 2,2p,3,3p,p+1 & \text{if } p\equiv 5 \pmod{12},\\ 2,2p,3,3p,p+2,2p+1 & \text{if } p\equiv 7 \pmod{12},\\ 2,2p,3,3p & \text{if } p\equiv 11 \pmod{12}. \end{cases}$$

The stated indices are known to be sufficient, while necessity remains open in general; the cases $p=5$ and $p=7$ are proved in the paper.