Local trajectory reconstruction conjecture for sorting networks

Let $\sigma^n$ be an $n$-element uniform random sorting network, let $J_n$ be uniform on ${1,\mathellipsis,n}$, and let $S_t^n(J_n,\cdot)$ be the unique curve of the form $A\sin(\pi s+\Theta)$ satisfying

$$S_t^n(J_n,0)=\sigma_G^n(J_n,0),\qquad S_t^n(J_n,t)=\sigma_G^n(J_n,t).$$

Local trajectory reconstruction conjecture. For every $\epsilon>0$, there exists $C>0$ such that

$$\liminf_{n\to\infty}\mathbb{P}\left(\|\sigma_G^n(J_n,\cdot)-S_{C/n}^n(J_n,\cdot)\|_u<\epsilon\right)\ge1-\epsilon.$$

This predicts that observing a randomly chosen particle after only order $n$ steps determines its full trajectory with high probability; the paper presents it as an open problem.