Schinzel's multivariate pullback factorization conjecture

Let $H\in\mathbb{C}[{\boldsymbol{y}}^{\pm1}]$ and let $k\geq 2$. For $A\in\mathbb{Z}^{n\times k}$, write $H({\boldsymbol{t}}^A)$ for the pullback of $H$ under the monomial map ${\boldsymbol{t}}\mapsto{\boldsymbol{t}}^A$, and let $S$ be the multiplicative subset generated by all $f({\boldsymbol{t}}^{\boldsymbol{d}})$ with $f\in\mathbb{C}[z^{\pm1}]$ and ${\boldsymbol{d}}\in\mathbb{Z}^k$.

Multivariate pullback factorization conjecture. There is a finite set of matrices $\Omega\subset\mathbb{Z}^{n\times n}$ such that, for each $A\in\mathbb{Z}^{n\times k}$, there are $N\in\Omega$ and $B\in\mathbb{Z}^{n\times k}$ with $A=NB$ such that if $P$ is an irreducible factor of $H({\boldsymbol{y}}^N)$, then $P({\boldsymbol{t}}^B)$ is, as an element of $\mathbb{C}[{\boldsymbol{t}}^{\pm1}]_S$, either a unit or an irreducible factor of $H({\boldsymbol{t}}^A)$.

This conjecture would give a finite description of irreducible factors that genuinely depend on more than one variable; the remaining univariate factors become units after localization. Its status is not resolved in the supplied text.