Archimedes configuration conjecture for uniform sorting networks

Let $\omega_n$ be an $n$-element uniform sorting network. For $t\in(0,1)$, define the scaled configuration by

$$\mu_t(\omega_n):=\frac 1n \sum_{i=1}^n \delta\Big(\frac {2i}{n}-1\;,\;\frac {2\sigma_{\lfloor tN\rfloor}(i)}{n}-1\Big).$$

For $t\in(0,1)$, define the Archimedes measure by

$$\mathfrak{Arch}_t(dx\times dy):=\frac{1}{2\pi}\sqrt{\Big[\sin^2(\pi t) +2xy\cos(\pi t)-x^2-y^2\Big]^{-1}\vee 0}\;dx\;dy.$$

Archimedes configurations. For all $t\in(0,1)$, the scaled configuration converges in distribution to the Archimedes measure in the vague topology for random Borel measures on $\mathbb{R}^2$:

$$\mu_t(\omega_n) \Longrightarrow \mathfrak{Arch}_t \qquad\text{as}\quad n\to\infty.$$

The conjecture describes the limiting particle density inside the asymptotic ellipses suggested by simulations. The supplied text gives no resolution evidence.