Corwin–Sheffield conjecture on extremal parabolic Brownian Gibbs line ensembles

Let $\mathcal{L} = (\mathcal{L}_1, \mathcal{L}_2, \ldots)$ be a line ensemble such that $\mathcal{L}(x)+x^2$ is stationary under deterministic horizontal shifts, $\mathcal{L}$ has the Brownian Gibbs property, and the law of $\mathcal{L}$ is extremal in the set of such Gibbs measures. Corwin–Sheffield conjecture. Then $\mathcal{L}$ is the parabolic Airy line ensemble, up to a trivial deterministic vertical shift of the entire ensemble. This conjecture concerns the classification of extremal Gibbs measures for line ensembles and appears in work of Corwin and Sheffield. Its resolution is not established by the supplied source.