Bucko–Czap conjecture on 1-planarity of lexicographic products

Let $G$ be a graph, and let $K_2$ denote the complete graph with two vertices. A graph is 1-planar if it has a drawing in the plane in which every edge is crossed by at most one other edge. A cactus is a connected graph in which every edge belongs to at most one cycle. Bucko–Czap's conjecture. The graph $G\circ K_2$ is 1-planar if and only if $G$ is a cactus. This open problem characterizes 1-planarity for lexicographic products with $K_2$ and concerns the relationship between graph substitution and sparse crossing structures.