Irreducibility and divisibility conjecture for the polynomials $C_n(x)$

Let $C_n(x)$ be the polynomial defined by the recurrence in the paper, with degree $n-1$.

Polynomial conjecture. The polynomials satisfy both of the following conditions:

$$\text{If }n\text{ is prime, then }C_n(x)\text{ is irreducible.}$$ $$C_m(x)\mid C_n(x)\quad\text{if and only if}\quad m\mid n.$$

These arithmetic properties are intended to describe the factorization pattern of the polynomials arising from the cusp-shape calculations for fully augmented pretzel links. The supplied text gives no resolution of either assertion.