The no-product-structure conjecture below the independent-crossing threshold

Let $n$ be a positive integer, let $s(n)$ be the threshold defined in the paper, and let $\alpha$ be a parameter. Consider the class of intersection graphs of $\alpha$-free homothetic regular $n$-gons.

The no-product-structure conjecture. For every $\alpha < s(n)$, the class of intersection graphs of $\alpha$-free homothetic regular $n$-gons does not have product structure.

The paper determines that arbitrarily many independent edges can cross a single edge in canonical drawings when $\alpha<s(n)$, but the required construction ruling out product structure is known only in some cases and an alternative construction is suspected for the remaining cases.