The product-structure conjecture above 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 above-threshold product-structure conjecture. For every $\alpha \geq s(n)$, the class of intersection graphs of $\alpha$-free homothetic regular $n$-gons has product structure.

For $\alpha\geq s(n)$, the paper proves that the canonical drawings are $k$-independent crossing with $k=26(n+1)$. The assertion would follow from the conjectured product structure for all $k$-independent crossing graph classes.