Reversed Dickson second-kind permutation classification for primes greater than seven

Let $p>7$ be a prime, and let $E_n(1,x)$ denote the reversed Dickson polynomial of the second kind. A permutation polynomial (PP) is a polynomial that permutes $\mathbb{F}_p$. The large-prime reversed Dickson permutation conjecture. $E_n(1,x)$ is a PP of $\mathbb{F}_p$ if and only if

$$n\equiv 2,3\pmod{p(p^2-1)}.$$

This extends the explicitly conjectured classifications for $p=5$ and $p=7$; the general classification for $p>7$ is not proved in the source.