Asymptotic gap conjecture for the KPZ line ensemble

Let $t>0$ and let $\mathcal{H}^{(t)}=\{\mathcal{H}^{(t)}_n\}_{n=1}^{\infty}$ denote the KPZ line ensemble at parameter $t$. The value $\mathcal{H}^{(t)}_n(0)$ is the height of its $n$th line at $0$.

Asymptotic gap conjecture. Almost surely as $n\to\infty$,

$$\mathcal{H}^{(t)}_{n+1}(0)-\mathcal{H}^{(t)}_n(0)=\log(n/t)+o(1).$$

This predicts that the KPZ line ensemble is eventually totally ordered, with consecutive gaps matching the logarithmic spacing suggested by the Toda locations. The statement is presented as an open problem, and the paper gives only heuristic support for the $o(1)$ error.