The crossing-number equality conjecture 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 $n_1,n_2,n_3$. Write $\operatorname{cr}(G)$ for the crossing number of a graph $G$, and $\overline{\operatorname{cr}}(G)$ for its rectilinear crossing number, the minimum number of crossings in a straight-line drawing of $G$.

Crossing-number equality conjecture.

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

Since every rectilinear drawing is a planar drawing, the left-hand side is always at least the ordinary crossing number. The conjecture asserts that complete tripartite graphs always admit optimal straight-line drawings; the supplied context gives evidence from the preceding bounds but no resolution.