Exponential concentration conjecture for sorting-network permutation measures

Let $\mathfrak{Arch}_t$ be the Archimedean path of probability measures on $[-1,1]^2$, and let $\eta_t^n$ be the time-$t$ permutation-matrix measure of an $n$-element uniform random sorting network. Let $U$ be an open set in the space of probability measures on $[-1,1]^2$ with the weak topology, containing every $\mathfrak{Arch}_t$.

Exponential concentration conjecture. There exist constants $c_1,c_2>0$ such that, for every $n$,

$$\mathbb{P}\left(\text{there exists }t\in[0,1]\text{ such that }\eta_t^n\notin U\right)\le c_1e^{-c_2n^2}.$$

This strengthens convergence to the Archimedean path from a fixed-time statement to uniform-in-time exponential concentration; the paper presents it as an open problem.