Lara Rodríguez–Thakur rationality conjecture for the ratios $\alpha_{n,1}$

For each of the four class-number-one base rings $A$ in Examples (i)–(iv), define

$$\alpha_{n,1}:=\frac{\zeta_A(q^n-1,q^{n+1}-q^n)}{\zeta_A(q^{n+1}-1)}.$$

The source introduces the corresponding variables $X$ and $Y$ as powers of $\theta$ and $\eta$ in each example. Lara Rodríguez–Thakur rationality conjecture. The ratios $\alpha_{n,1}$ belong to $K$ in all four cases. The displayed formulas that follow establish explicit expressions for these ratios in the four examples, providing the claimed rationality in those cases; the conjectural origin is Lara Rodríguez and Thakur's earlier conjecture, and the source gives no further unresolved qualification.