Chebyshev–Artin primitive-root conjecture

For a prime $p$, let $R_p$ denote the relevant Chebyshev residue structure, and let $\omega_a$ be the element associated with $a\in R_p$. A Chebyshev primitive root is an element $a\in R_p$ for which $\omega_a$ has order $p-\epsilon$. A Chebyshev square is an integer of the form $T_2(x)=2x^2-1$. Chebyshev–Artin primitive-root conjecture. Every integer $a\neq 0,\pm1,2n^2+1$ that is not a Chebyshev square is a Chebyshev primitive root modulo $p$ for infinitely many primes $p$. This is the proposed analogue of Artin's conjecture for the Chebyshev setting; the source gives no resolution, so the assertion remains open.