A congruence holding for primes $p \equiv 1,11 \pmod {12}$

Question

Can you provide a proof or counterexample to the following claim ?

Let $p$ be prime number greater than three and let $T_n(x)$ be Chebyshev polynomial of the first kind , then $T_{p-1}(2) \equiv 1 \pmod p$ if and only if $p \equiv 1,11 \pmod {12}$ .

I have tested this claim up to $2 \cdot 10^6$ .

I was searching for a counterexample using the following two PARI/GP codes :

FermatChebyshev1(lb,ub)={
forprime(p=lb,ub,
if(Mod(p,12)==1 || Mod(p,12)==11 ,
if(!(Mod(polchebyshev(p-1,1,2),p)==1),print(p))))
}

FermatChebyshev2(lb,ub)={
forprime(p=lb,ub,
if(!(Mod(p,12)==1 || Mod(p,12)==11) ,
if(Mod(polchebyshev(p-1,1,2),p)==1,print(p))))
}