Prime-value conjecture for the periodic-continued-fraction polynomials

Let $f_n(x)$ be the polynomial sequence defined earlier in the paper, and let $a$ be a positive integer. Prime-value conjecture. For every positive integer $a$, there are infinitely many $n\in\mathbb{N}$ such that $|f_n(a)|$ is a prime number. If true, this would show that the condition on the exponent of $p$ in the second part of the paper's cited theorem cannot be dropped. The assertion is presented as an open conjecture and is distinct from the irreducibility conjecture, since it fixes $a$ and varies $n$.