Infinitely many primes in the companion sequence $r_k(n)$

Let $(r_k(n))_{k\geq 0}$ be the companion sequence associated with the positive integer parameter $n$, and let $\mathcal{T}_p(j)$ denote the relevant dilated Chebyshev value. Prime-occurrence conjecture for $r_k(n)$. For every positive integer $n>2$, the sequence

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

contains infinitely many primes if and only if $n\neq\mathcal{T}_p(j)$ for every prime $p$, where the integer $j\geq 3$ is restricted to one of the values specified in the paper's Theorem $\mathrm{tmaint}$. This is presented as the analogous expected behaviour for the companion family; no proof or resolution is supplied.