The rectilinear crossing-number formula for complete tripartite graphs

Let $n_1,n_2,n_3\geq 1$, and let $K_{n_1,n_2,n_3}$ be the complete tripartite graph with part sizes $n_1,n_2,n_3$. Its rectilinear crossing number, denoted by $\overline{\operatorname{cr}}(K_{n_1,n_2,n_3})$, is the minimum number of crossings in a drawing in which every edge is a straight line segment. Let $A(n_1,n_2,n_3)$ be the upper-bound expression defined in the paper.

Rectilinear crossing-number conjecture.

$$\overline{\operatorname{cr}}(K_{n_1,n_2,n_3})=A(n_1,n_2,n_3).$$

The preceding upper and lower bounds provide evidence for this formula, which is known in several cases where two parts are small but remains open in general.