Sheffield's uniqueness conjecture for extremal Brownian Gibbs line ensembles

Let $\mathcal{A}$ be an $\mathbb{N}$-indexed line ensemble. Say that $\mathcal{A}$ is $x$-invariant if $\mathcal{A}(s+\cdot)$ has the same distribution as $\mathcal{A}$ for every $s\binmathbb{R}$. Let $\mathcal{L}$ denote the Airy line ensemble from Theorem~, and define

$$\mathcal{A}_i(t)=2^{1/2}\mathcal{L}_i(t)+t^2,$$

for $i\in\mathbb{N}$. Sheffield's uniqueness conjecture. The set of extremal Brownian Gibbs $\mathbb{N}$-indexed line ensembles $\mathcal{L}$ for which the associated ensemble $\mathcal{A}$ is $x$-invariant is exactly

$$\big\{\mathcal{L}^{(0,y)}:y\in\mathbb{R}\big\}.$$

The conjecture proposes that the vertically shifted Airy line ensembles exhaust the extremal Brownian Gibbs measures with this translation-invariance property. The source attributes the possibility to Scott Sheffield and does not state a resolution, so the conjecture is recorded as open.