Cyclotomic root conjecture for the inversion polynomial

Let $b=ck$, and let $\zeta_m$ denote a primitive $m$th root of unity. The inversion polynomial $f_b(x)$ is defined by its roots indexed by the inversion numbers $\inv(a,b)$. Cyclotomic root conjecture. The numbers $\zeta_{2k}$ and $\zeta_{6k}$ are roots of $f_b(x)$ precisely under the following conditions: (i) If $c=2 \pmod{4}$, then both are roots precisely when $k=4 \pmod{8}$. (ii) If $c=0 \pmod{4}$, then both are roots precisely when $k \neq 2 \pmod{4}$ and $8\nmid k$. (iii) If $c=3^m n$, $m\geq1$, and $(n,6)=1$, then only $\zeta_{2k}$ is a root precisely when $3n\nmid k$ and $c$ is not a square. (iv) If $(c,6)=1$, then both are roots precisely when $c\nmid k$ and $c$ is not a square. This conjecture characterizes most of the cyclotomic factors of $f_b(x)$; several parts are proved in the paper, while the stated combined characterization remains a conjecture.