Archimedean limit conjecture for sorting-network permutation matrices

Let $\sigma^n$ be an $n$-element uniform random sorting network, and let $\sigma_G^n(i,t)$ be its globally rescaled particle positions. Let $\mathfrak{Arch}_{1/2}$ be the probability measure on $[-1,1]^2$ with density $1/(2\pi\sqrt{1-x^2-y^2})$ on the unit disk, and let $\mathfrak{Arch}_t$ be the distribution of $(X,X\cos(\pi t)+Y\sin(\pi t))$ when $(X,Y)$ has law $\mathfrak{Arch}_{1/2}$. Define

$$\eta_t^n=\frac1n\sum_{i=1}^ndelta(\sigma_G^n(i,0),\sigma_G^n(i,t)).$$

Angel–Holroyd–Romik–Virág Archimedean-limit conjecture. For every $t\in[0,1]$,

$$\eta_t^n\tomathfrak{Arch}_t$$

in probability in the weak topology. Equivalently, for every weakly open neighbourhood $O$ of $\mathfrak{Arch}_t$, $\mathbb{P}(\eta_t^n\in O)\to1$ as $n\to\infty$. This conjecture gives the global limiting shape of time-$t$ permutation matrices; the source says that subsequent work proves it.