Reversed Dickson second-kind permutation classification over the field of five elements

Let $E_n(1,x)$ denote the reversed Dickson polynomial of the second kind, and let a permutation polynomial (PP) be a polynomial that permutes the field. The $p=5$ reversed Dickson permutation conjecture. Over $\mathbb{F}_5$, $E_n(1,x)$ is a PP if and only if

$$n\equiv 2,3,15,94\pmod{120}.$$

The listed residue classes are verified in the paper for the displayed representatives, but the claimed complete classification is stated as a conjecture.