The asymptotic Zarankiewicz conjecture for crossing numbers

Let $cr(K_{n,n})$ be the crossing number of the complete bipartite graph $K_{n,n}$, and let $cr_0(K_{n,n})$ be its geodesic crossing number. Define

$$\lambda:= \lim_{n\to\infty} n^{-4} cr(K_{n,n})$$

and

$$\lambda_0:= \lim_{n\to\infty} n^{-4} cr_0(K_{n,n}).$$

The limits exist, and satisfy $\lambda \leq \lambda_0 \leq \tfrac{1}{16}$. Asymptotic Zarankiewicz conjecture.

$$\lambda = \lambda_0 = \tfrac{1}{16}.$$

This conjecture asserts that the Zarankiewicz construction is asymptotically optimal both for the usual crossing number and for the geodesic crossing number. The source states that the conjecture remains open.