Great-circle conjecture for uniform sorting networks

Let $\omega_n$ be an $n$-element uniform sorting network, let ${\mathbb S}_n$ denote the permutahedron, and let $d_\infty$ be the distance from a sorting network to a great circle defined by

$$d_\infty(\omega,c):=\max_{i\in[1,N]}\inf_{z\in c}\|\sigma_i^{-1}-z\|_\infty.$$

A sequence is $o(n)$ in probability if its ratio to $n$ converges to zero in probability. Great circles. For each $n$ there exists a random great circle $C_n\subset{\mathbb S}_n$ such that

$$d_\infty(\omega_n,C_n)=o(n)\quad\text{in probability as}\quad n\to\infty.$$

If true, this would explain the conjectured sine trajectories, Archimedes limiting configurations, and semicircle-law swap process through the deterministic great-circle theorem stated nearby. The supplied text does not resolve whether uniform sorting networks lie close to random great circles.