Adjacency distribution conjecture for random sorting networks

Let $Y_n(\alpha)$ and $X_n(\alpha)$ denote the number of adjacencies $1\leq k<\alpha\binom{n}{2}$ in a random sorting network and in a random $132$-avoiding sorting network of size $n$, respectively. Here $\alpha\in[0,1]$, and let $g:[0,1]\to\mathbb{R}$ be defined by

$$g(\alpha)= \begin{cases} \sqrt{\frac{\alpha}{2}}&\quad\text{if }\alpha\in[0,\frac{1}{2}],\\ 1-\sqrt{\frac{1-\alpha}{2}}&\quad\text{if }\alpha\in[\frac{1}{2},1]. \end{cases}$$

Adjacency distribution conjecture. The random sorting-network statistic satisfies $Y_n(\alpha)/\mathbb{E}[Y_n(1)]\to c\alpha$ in probability for some constant $c$, and the random $132$-avoiding sorting-network statistic satisfies

$$\lim_{n\to\infty}\mathbb{P}\left(\max_{0\leq\alpha\leq 1}\left|\frac{X_n(\alpha)}{2(n-2)}-g(\alpha)\right|>\epsilon\right)=0.$$

The claim predicts linear growth of the normalized adjacency count in random sorting networks and a deterministic, piecewise-square-root limiting profile in random $132$-avoiding sorting networks. The surrounding discussion says that these behaviors are suggested by experiments; no resolution is given here.