Parabolic rigidity conjecture for the KPZ line ensemble

Fix $t>0$, and let $\mathcal{H}^{(t)}=\{\mathcal{H}^{(t)}_n\}_{n=1}^{\infty}$ be the KPZ line ensemble at parameter $t$. For each $n$, regard $\mathcal{H}^{(t)}_n-\mathcal{H}^{(t)}_n(0)$ as a random function of $x$.

Parabolic rigidity conjecture. As $n\to\infty$, the sequence of functions

$$\mathcal{H}^{(t)}_n-\mathcal{H}^{(t)}_n(0)$$

converges almost surely, uniformly on compact sets, to the deterministic parabola $-x^2/(2t)$.

This predicts that the individual KPZ lines become rigid after recentering at their values at the origin. It is stated among the paper's open problems and is motivated heuristically by Brownian resampling and the large-index Toda locations.