Can you prove or disprove the following claim:

Let $N=4p+1$ where $p$ is an odd prime number , let $T_n(x)$ be the nth Chebyshev polynomial of the first kind and let $F_n(x)$ denote an irreducible factor of degree $\varphi(n)$ of $T_n(x)$ . If there exists an integer $a$ such that $F_{p}(a) \equiv 0 \pmod{N} $ then $N$ is a prime.

You can run this test here. An incomplete list of primes $p$ such that $4p+1$ is prime can be found here . I have verified this claim for $p \in [3,30000)$ with $a \in [1,100]$ .