Maximum fluctuation conjecture for halfway permutation paths

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

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

Maximum fluctuation conjecture. For every $\epsilon>0$,

$$\max_{j\in[1,n]}\sup_{s,t\in[0,1]}n^{1/2-\epsilon}|Z_j^n(t)-Z_j^n(s)|\to0$$

in probability as $n\to\infty$. This predicts a uniform upper bound, up to an arbitrary power $n^\epsilon$, on fluctuations of all halfway-permutation paths; the paper presents it as an open problem.