I am interested in the status of the conjecture about the minimum number of edge crossings $cr(K_{m,n})$ in a drawing of the complete bipartite graph $K_{m,n}$.

The Wikipedia article https://en.wikipedia.org/wiki/Tur%C3%A1n%27s_brick_factory_problem led me to study the original papers of Zarankiewicz (*On a problem of P. Turan concerning graphs*) from 1954 and of Urbanik (*Solution du problème posé par P. Turán*) from 1955.

I wondered whether someone could tell me whether an asymptotic approach has been successfully attempted (letting $n\to\infty$). If so, I would be very interested in any references for that.