Parker's factorization and palindromicity conjecture for cyclotomic coordinator polynomials

Let $m$ be a positive integer. Write $\sqrt m$ for the squarefree part of $m$, let $\zeta_m=e^{2\pi i/m}$, and let $h_m(x)$ be the coordinator polynomial of the lattice $\mathbb Z[\zeta_m]$ with respect to the set of all $m$th roots of unity. A degree-$d$ polynomial $c_dx^d+c_{d-1}x^{d-1}+\dots+c_0$ is palindromic when $c_k=c_{d-k}$. Parker's conjecture. The coordinator polynomial satisfies

$$h_m(x)=\left(h_{\sqrt m}(x)\right)^{m/\sqrt m},$$

and $h_{\sqrt m}(x)$ is a palindromic polynomial of degree $\phi(\sqrt m)$. This conjecture is partly proved in the paper: the factorization holds for every positive integer $m$, while the palindromicity assertion is established for several families but remains open in general.