Zarankiewicz's conjecture for complete bipartite graphs

Let $K_{m,n}$ be the complete bipartite graph with part sizes $m$ and $n$, and define

$$Z(m,n):=\left\lfloor\frac{n}{2}\right\rfloor\left\lfloor\frac{n-1}{2}\right\rfloor\left\lfloor\frac{m}{2}\right\rfloor\left\lfloor\frac{m-1}{2}\right\rfloor.$$

Here $\operatorname{cr}(G)$ denotes the minimum number of crossings in a plane drawing of a graph $G$. Zarankiewicz conjecture.

$$\operatorname{cr}(K_{m,n})=Z(m,n).$$

Zarankiewicz constructed a drawing attaining this number, but a flaw was found in his claimed proof of optimality. The conjecture remains open in general.