Can you provide a proof or a counterexample for the claim given below ?

Inspired by Agrawal's conjecture in this paper and by Theorem 4 in this paper I have formulated the following claim :

Let $n$ be a natural number greater than two . Let $r$ be the smallest odd prime number such that $r \nmid n$ and $n^2 \not\equiv 1 \pmod r$ . Let $T_n(x)$ be Chebyshev polynomial of the first kind , then $n$ is a prime number if and only if $T_n(x) \equiv x^n \pmod {x^r-1,n}$ .

You can run this test here .

I have tested this claim up to $5 \cdot 10^4$ and there were no counterexamples .