The product-structure conjecture for k-independent crossing graphs

Let $k$ be a nonnegative integer. A graph is $k$-independent crossing if it has a topological drawing in which no edge is crossed by more than $k$ independent edges.

Product-structure conjecture for $k$-independent crossing graphs. The class of $k$-independent crossing graphs admits product structure.

This would extend the known product-structure results for planar and other beyond-planar graph classes. The paper applies the conjecture to canonical drawings of regular polygon intersection graphs, where bounded independent crossing is proved for $\alpha\geq s(n)$, but the general product-structure assertion remains open.