The multiples-of-factorial conjecture for Ehrhart-equivalent integral polytopes

Let $n\in\mathbb{Z}_{>3}$, and let $P,Q\subset\mathbb{R}^n$ be arbitrary Ehrhart-equivalent integral $n$-polytopes. For a positive integer $j$, the notation $jP$ denotes the dilation of $P$ by $j$, and $\operatorname{GL}_n(\mathbb{Z})$-equidecomposability is by affine-unimodular transformations.

The multiples-of-factorial conjecture. For any $n\in\mathbb{Z}_{>3}$, if $P,Q\subset\mathbb{R}^n$ are two arbitrary Ehrhart-equivalent integral $n$-polytopes, then

$$(j(n-1)!)P\text{ and }(j(n-1)!)Q$$

are $\operatorname{GL}_n(\mathbb{Z})$-equidecomposable for all $j\in\mathbb{Z}_{\ge 1}$.

This conjecture proposes an infinite class of dilation factors beyond the single factor $(n-1)!$. The paper notes that infinitely many dilation factors exist in general, while for $n=1$ and $n=2$ every positive integer works; the stated conjecture concerns dimensions greater than three.