The irreducibility conjecture for normalized cusp polynomials

Define

$$P_d(t;q):= \begin{cases} \mathit{NH}_d(t;q),&d\text{ even},\\ \mathit{NH}_d(t;q)/(1+q^d t),&d\text{ odd}. \end{cases}$$

Let $\Phi_r(q)$ denote the cyclotomic polynomial of order $r$. The normalized cusp-polynomial irreducibility conjecture. The polynomial $P_d(t;q)$ lies in $\mathbb{Z}[t,q]$, has nonnegative coefficients, and is irreducible or equal to $1$. For every integer $m\ne-1$, $P_d(m;q)$ is irreducible in $\mathbb{Z}[q]$. Finally,

$$P_d(-1;q)=I_d(q)\prod_{\substack{1\leq r\leq d\r\text{ odd}}}\Phi_r(q)^{\left\lfloor\frac{d+r-1}{2r}\right\rfloor},$$

where $I_d(q)$ is irreducible in $\mathbb{Z}[q]$.

The factorization pattern was verified computationally for $d\leq 30$ and selected values of $t$, but the general assertions remain open.