Permutation-process convergence to the Archimedean process

Let $\mathbf{RSN}^n$ be a uniformly random sorting network of $\mathcal{S}_n$, regarded as a permutation process, and let $\mathcal{A}=(\mathcal{A}(t);0\leq t\leq1)$ be the Archimedean process defined by $\mathcal{A}(t)=\cos(\pi t)\mathbf{A}_x+\sin(\pi t)\mathbf{A}_y$, with $(\mathbf{A}_x,\mathbf{A}_y)$ distributed according to the Archimedean measure.

Archimedean process convergence conjecture. $\mathbf{RSN}^n$ converges in probability, as a permutation process, to the Archimedean process.

This conjecture combines the expected Archimedean path and sine-curve behavior into a process-level convergence statement. The source presents it as a reasonable conjecture and does not report a resolution.