Complete permutation classification for reversed Dickson second-kind polynomials

Let $p\geq 5$ be a prime, and let $E_n(1,x)$ denote the reversed Dickson polynomial of the second kind. A complete permutation polynomial (CPP) is a polynomial $f$ such that both $f(x)$ and $f(x)+x$ permute $\mathbb{F}_p$. Reversed Dickson second-kind complete permutation conjecture. $E_n(1,x)$ is a CPP of $\mathbb{F}_p$ if and only if

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

This is the complete-mapping analogue of the preceding permutation conjecture and is presented without a proof; its general validity remains open.