Infinitely many primes in the Chebyshev-related sequence $s_k(n)$

Let $(s_k(n))_{k\geq 0}$ be the sequence associated with the positive integer parameter $n$, and let $\mathcal{T}_p(j)$ denote the dilated Chebyshev polynomial of the first kind evaluated at the integer $j$. Prime-occurrence conjecture for $s_k(n)$. For every positive integer $n>1$, the sequence

$$(s_k(n))_{k\geq 0}$$

contains infinitely many primes if and only if $n\neq\mathcal{T}_p(j)$ for every prime $p$ and every integer $j\geq 3$. The exceptional Chebyshev values are known to yield at most one prime term, while the assertion of infinitely many primes for all other values remains open.