Strict crossing-number separation for balanced complete 4-partite graphs

Let $K_n^4$ denote the complete balanced $4$-partite graph with $n$ vertices, and let $\operatorname{cr}(G)$ and $\overline{\operatorname{cr}}(G)$ denote respectively its ordinary and rectilinear crossing numbers. 4-partite separation conjecture. There exists a natural number $n_0>9$ such that, for every $n\geq n_0$,

$$\operatorname{cr}(K_n^4)<\overline{\operatorname{cr}}(K_n^4).$$

The known values and upper bounds at $n=8,9$ motivate the conjecture that, eventually, straight-line drawings require strictly more crossings than unrestricted drawings. The source does not determine such an $n_0$.