The conjecture on the polynomial part of the invariant ideal

Let $n\ge3$ and let $I_n=\langle p_n,q_n\rangle\subset\mathbf Z[z,t]$, with $p_n$ and $q_n$ as defined in the paper. The intersection $I_n\cap\mathbf Z[t]$ is an ideal of $\mathbf Z[t]$. The polynomial-part ideal conjecture. The ideal $I_n\cap\mathbf Z[t]$ is principal and is generated by

$$(1-t^2)\prod_{i=1}^{n-1}(1-t^{2i}),$$

when $n$ is odd; by

$$(1+t)\prod_{i=1}^{n-1}(1-t^i),$$

when $n$ is congruent to $2$ modulo $4$; and by

$$\prod_{i=1}^{n-1}(1-t^i),$$

when $n$ is divisible by $4$. The paper calls this an intriguing conjecture and notes that it plays no role in the algorithm.