Airy-type characterization conjecture for the KPZ line ensemble

Let $\mathcal{L}=\{\mathcal{L}_n\}_{n=1}^{\infty}$ be a collection of random curves with the $\mathbf{H}$-Brownian Gibbs property for $\mathbf{H}(x)=e^x$. Assume that for every $\epsilon>0$ there is a constant $\kappa(\epsilon)>0$ such that, for all $x\in\mathbb{R}$,

$$\mathbb{P}\left(\left|\mathcal{L}_1(x)-\frac{x^2}{2t}\right|\leq\epsilon x^2+\kappa(\epsilon)\right)\geq 1-\epsilon.$$

Airy-type characterization conjecture. Then $\mathcal{L}$ is equal in distribution to the $\mathrm{KPZ}_t$ line ensemble, up to an independent affine shift.

This is proposed as an analogue of the characterization of the Airy line ensemble. The source gives no resolution and explicitly presents it as a conjectural endpoint for a characterization theory; its precise relation to the cited Airy characterization should be checked.