Czap, Przybyło and Škrabul'áková's bipartite 1-planar graph size conjecture

Let $G$ be a bipartite $1$-planar graph with partite sets $X$ and $Y$, where $|X|$ and $|Y|$ are integers satisfying

$$|X|\geq 3\quad\text{and}\quad |Y|\geq 6|X|-12.$$

Czap, Przybyło and Škrabul'áková's conjecture. The size of $G$ satisfies

$$|E(G)|\leq 2|V(G)|+4|X|-12.$$

The conjecture proposes that the lower bound constructed by Czap, Przybyło and Škrabul'áková is optimal when the partite sets have these sizes. The claim concerns the extremal size of bipartite $1$-planar graphs with prescribed partite-set sizes and remains unresolved in the supplied source.