The superlinear crossing-family conjecture for geometric complete graphs

Let $t(n)$ be the minimum number of colours needed to colour the edges of some geometric drawing of $K_n$ so that every pair of edges with the same colour crosses; each colour class is a crossing family. Crossing-family conjecture. The colouring number satisfies

$$\frac{t(n)}{n}\rightarrow\infty\quad\text{as }n\rightarrow\infty.$$

Since each crossing family can be associated with blockers, this is stated as a strengthening of the superlinear blocking conjecture and remains open.