The crossing-number conjecture for the crown graph

Let $K_{n,n}-nK_2$ be the bipartite graph obtained from the complete bipartite graph $K_{n,n}$ by deleting a perfect matching, and let $cr(G)$ denote the crossing number of a graph $G$.

Crossing-number conjecture.

$$cr(K_{n,n}-nK_2)=\left\lfloor\frac{n}{2}\right\rfloor\left\lfloor\frac{n-1}{2}\right\rfloor\left\lfloor\frac{n-2}{2}\right\rfloor\left\lfloor\frac{n-3}{2}\right\rfloor.$$

This conjecture gives an exact formula for the crossing number of the crown graph. The paper establishes matching upper and substantially weaker lower bounds in general, and proves the formula for $n=5$; the general equality remains open in the supplied text.