Schinzel's non-cyclotomic factorization conjecture

Let $F\in\mathbb{Q}[{\boldsymbol{x}}^{\pm1}]$ be a non-cyclotomic irreducible Laurent polynomial. Let $\langle{\boldsymbol{c}},{\boldsymbol{a}}\rangle$ denote the scalar product of vectors in $\mathbb{Z}^n$, and let $\operatorname{cyc}(G)$ denote the cyclotomic part of a Laurent polynomial $G$.

Schinzel's non-cyclotomic factorization conjecture. There are finite sets $\Omega^0\subset\mathbb{Z}^{n\times n}$ of nonsingular matrices and $\Gamma\subset\mathbb{Z}^n$ of nonzero vectors such that, for every ${\boldsymbol{a}}\in\mathbb{Z}^n$, either there is ${\boldsymbol{c}}\in\Gamma$ with $\langle{\boldsymbol{c}},{\boldsymbol{a}}\rangle=0$, or there are $M\in\Omega^0$ and ${\boldsymbol{b}}\in\mathbb{Z}^n$ with ${\boldsymbol{a}}=M{\boldsymbol{b}}$ such that, if $F({\boldsymbol{x}}^M)=\prod_P P^{e_P}$ is its irreducible factorization, then

$$\frac{F(t^{\boldsymbol{a}})}{\operatorname{cyc}(F(t^{\boldsymbol{a}}))}=\prod_P\left(\frac{P(t^{\boldsymbol{b}})}{\operatorname{cyc}(P(t^{\boldsymbol{b}}))}\right)^{e_P}$$

is the irreducible factorization of $F(t^{\boldsymbol{a}})/\operatorname{cyc}(F(t^{\boldsymbol{a}}))$.

The conjecture gives a uniform factorization pattern away from finitely many orthogonality hyperplanes, after removing cyclotomic factors. The supplied text attributes it to Schinzel and does not resolve its status.