Fluctuation-variance conjecture for halfway permutation paths

Let $\sigma^n$ be an $n$-element uniform random sorting network and define

$$Z_j^n(t)=e^{\pi i t}\left[\sigma_G^n(j,t)+isigma_G^n(j,t+1/2)\right],\qquad t\in[0,1/2].$$

Let $U$ be uniform on $[0,1]$, independent of all the sorting networks, and let $J_n$ be uniform on ${1,\mathellipsis,n}$, independent of $\sigma^n$ and $U$.

Fluctuation-variance conjecture. The sequence

$$\left\{n\operatorname{Var}(Z_{J_n}^n(U)\mid J_n,\sigma^n):n\in\mathbb{N}\right\}$$

is tight, and there exist independent random variables $X_1,X_2$ such that

$$\left(n\operatorname{Var}(Z_{J_n}^n(U)\mid J_n,\sigma^n),|Z_{J_n}^n(0)|\right)\stackrel{d}{\to}\left(X_1\sqrt{1-X_2^2},X_2\right).$$

This conjecture refines the global limit by predicting the order and spatial dependence of fluctuations of localized halfway-permutation paths; the paper presents it as an open problem.