Period conjecture for reversed Dickson second-kind values at one quarter for non-Mersenne primes

Let $p>3$ be a non-Mersenne prime, and let $E_n(1,x)$ denote the reversed Dickson polynomial of the second kind. Consider the sequence of values $E_n(1,\frac14)$ modulo $p$. Non-Mersenne period conjecture. The sequence $\{E_n(1,\frac14)\}_{n=1}^{\infty}$ has period $p(p-1)$. The paper reports computational evidence for this periodicity; a proof is not supplied.