The sector cross-ratio supremum conjecture

Let $S_\theta$ be an open sector with angle $\pi<\theta<2\pi$. For $r>0$ and $0<k<1$, set

$$x=re^{(1-k)\theta i/2},\qquad y=re^{(1+k)\theta i/2}.$$

The sector cross-ratio supremum conjecture. One has

$$\sup_{a,b\in\partial S_\theta}|a,x,b,y|=\max\{|r,x,re^{\theta i},y|,|0,x,\infty,y|\}.$$

This is supported by computational experiments and concerns the extremal cross-ratio on a non-convex sector; no proof or resolution is given here.