Goh–Wildberger factorization conjecture for zpread polynomials

Let $Z_n(x)$ denote the $n$th zpread polynomial. For each integer $d\geq 1$, let $\Phi_d(x)\in\mathbb{Z}[x]$ have degree $\phi(d)$, where $\phi$ is Euler's totient function, and let $\mu$ denote the Möbius function.

Goh–Wildberger conjecture. There are polynomials $\Phi_d(x)\in\mathbb{Z}[x]$, $d\geq 1$, such that, for every $n\geq 1$,

$$Z_n(x)=\prod_{d\mid n}\Phi_d(x).$$

For $d\geq 3$, $\Phi_d(x)=\psi_d(x)^2$, where $\psi_d(x)\in\mathbb{Z}[x]$ is irreducible and has constant term $\psi_d(0)>0$ satisfying

$$\psi_d(0)=\prod_{d\mid n}\left(\frac{n}{d}\right)^{\mu(d)}.$$

Moreover, if $p\geq 5$ is prime, then

$$\psi_p(1)=(-1)^{\phi(p)/2}.$$

This conjecture reformulates a conjecture of Goh and Wildberger about factorizations of spread polynomials in terms of zpread polynomials, while also making explicit arithmetic patterns in the factors' coefficients. The cited OEIS entry records the proposed constant-term sequence; the supplied text gives no evidence resolving the conjecture.