The irreducibility conjecture for the numerator of the binary-form Poincaré series

Let $n\ge3$, and write the Poincaré series in lowest terms as

$$P_n(t)=\frac{A_n(t)}{B_n(t)},$$

where $A_n,B_n\in\mathbf Z[t]$ are relatively prime and $A_n(0)=B_n(0)=1$. Irreducibility conjecture. The polynomial $A_n(t)$ is irreducible for every $n\ge5$. The paper presents this as an additional conjecture after reporting computations through $n=30$; no general proof is given in the supplied text.