The canonical-drawing characterization of product structure for regular polygon intersection graphs

Let $n$ be a positive integer, let $\alpha$ be a parameter, and consider the class of intersection graphs of $\alpha$-free homothetic regular $n$-gons. A canonical drawing is a drawing associated with such an intersection representation, and a drawing is $k$-independent crossing if no edge is crossed by more than $k$ independent edges.

The canonical-drawing characterization. The class of intersection graphs of $\alpha$-free homothetic regular $n$-gons admits product structure if and only if their canonical drawings are $k$-independent crossing for a global constant $k$, possibly depending on $n$.

This conjecture proposes that product structure is exactly captured by bounded independent crossing in the canonical drawings; the paper establishes bounded independent crossing in one parameter regime but leaves the converse characterization open.