Period conjecture for reversed Dickson second-kind polynomials over finite rings

Let $p>3$ be a prime, let $n_1,n_2>1$, and let $E_n(1,x)$ denote the reversed Dickson polynomial of the second kind over $\mathbb{Z}_p$. Reversed Dickson second-kind period conjecture. If

$$n_1\equiv n_2\pmod{p(p^2-1)},$$

then

$$E_{n_1}(1,x)=E_{n_2}(1,x)$$

for every $x\in\mathbb{Z}_p$. The conjecture is presented as an immediate consequence of the two preceding period conjectures and remains unproved in the source.