Nonattainment conjecture for the rectilinear crossing density of the complete graphon

Let $W^1$ denote the graphon of the complete graph, let $\mathcal F$ be the family of graphon drawings used to define the rectilinear crossing density, and let $\overline{\mathop{\mathrm{cd}}}(W,f)$ be the corresponding density for $f\in\mathcal F$. Write $\overline{\mathop{\mathrm{cd}}}(W^1,\mathcal D)$ for the infimum over the admissible drawing family $\mathcal D$.

Nonattainment conjecture. There exists no $f\in\mathcal F$ such that

$$\overline{\mathop{\mathrm{cd}}}(W^1,\mathcal D)=\overline{\mathop{\mathrm{cd}}}(W^1,f).$$

In particular, the infimum in

$$\lim_{n\rightarrow\infty}\frac{\overline{\mathop{\mathrm{cr}}}(K_n)}{\binom{n}{4}}=\inf_{R}c(\lambda_R)$$

can not be substituted for a minimum. This concerns whether the limiting rectilinear crossing density of complete graphs is attained by a single admissible graphon drawing, and the source presents the attainment question as unresolved.