AHRV frozen-region limit-shape conjecture for random sorting networks

Let $w=w_1\dots w_{\binom{n}{2}}$ be a random sorting network, and fix $t\in(0,1)$. Define the scaled frozen region

$$F_t=\left\{\left(\frac{2j}{n}-1,1-\frac{2i}{n}\right)\in\mathbb{R}^2:(i,j)\in\lambda_f\left(w_1\dots w_{\left\lfloor t\binom{n}{2}\right\rfloor}\right)\right\}.$$

AHRV frozen-region conjecture. As $n\to\infty$, the boundary of the limit shape of $F_t$ is

$$\left\{(x,y)\in\mathbb{R}^2:x\leq-\cos(\pi t),\ y\geq\cos(\pi t),\ \sin^2(\pi t)-2xy\cos(\pi t)-x^2-y^2=0\right\}.$$

This is an informal reformulation of part of a conjecture of AHRV concerning the asymptotic geometry of random sorting networks and their frozen regions; the source does not provide evidence of a resolution.