Corwin–Hammond classification conjecture for extremal Brownian Gibbs line ensembles

A Brownian Gibbs $\mathbb{N}$-indexed line ensemble is a random collection of curves $\mathcal{L}=(\mathcal{L}_i)_{i\in\mathbb{N}}$ satisfying the Brownian Gibbs property. Define the parabolically shifted ensemble $\mathcal{A}$ by

$$\mathcal{A}_i(t)=2^{1/2}\mathcal{L}_i(t)+t^2,\qquad i\in\mathbb{N}.$$

An ensemble is extremal if it is an extremal point in the convex set of Brownian Gibbs line ensembles, and it is horizontal shift-invariant if its law is invariant under horizontal translations of the time parameter. Corwin–Hammond's conjecture. The set of extremal Brownian Gibbs $\mathbb{N}$-indexed line ensembles $\mathcal{L}$ for which $\mathcal{A}$ is horizontal shift-invariant is

$$\left\{\mathcal{L}^{\mathrm{Airy}}+y:y\in\mathbb{R}\right\},$$

where $\mathcal{L}^{\mathrm{Airy}}$ denotes the Airy line ensemble. This is a classification problem for stationary Brownian Gibbs line ensembles; the Airy line ensemble provides the known model, while the conjecture asserts that its vertical translates are the only extremal examples.