I have an even number of points $n$ randomly distributed (uniformly) in a disk. Then the points are randomly connected to form $n/2$ segments, a perfect matching. Finally, I form the intersection graph $G$ whose nodes are the segments, and each of whose edges represents a proper intersection between a pair of segments. An example is shown below:

My question is:

. What is theQdiameterof $G$ as $n \to \infty$?

I have some evidence that the diameter approaches a constant $> 1$, perhaps $4$ or $5$, but to be honest, the evidence is not robust.

**This may not be helpful, but for $n=1188$, $n_{\mathop{segs}}=n/2=594$, one simulation resulted in $G$ having $37,117$ edges, mean vertex degree $125$, one connected component, and diameter $4$:**

*Added.*