Sheffield's extremal Brownian Gibbs line ensemble conjecture

Let $\Theta=\{\theta_s\mid s\in\mathbb{R}\}$ be the horizontal shift group of $\mathbb{N}\times\mathbb{R}$-indexed line ensembles, acting by

$$\theta_s\mathcal{L}_i(x)=\mathcal{L}_i(x+s)\qquad\text{for all }i\in\mathbb{N},\ x\in\mathbb{R}.$$

A line ensemble is horizontal shift-invariant if $\theta_s\mathcal{L}$ has the same distribution as $\mathcal{L}$ for every $s\in\mathbb{R}$. Let $\mathscr{G}_{\Theta}$ be the set of Brownian Gibbs line ensemble measures $\mathcal{L}$ such that $2^{1/2}\mathcal{L}_i(x)+x^2$ is horizontal shift-invariant. Let $\mathcal{A}$ be the Airy line ensemble and define

$$\mathcal{L}^c_i(x)=\frac{1}{\sqrt{2}}\bigl(\mathcal{A}_i(x)-x^2\bigr)+c.$$

Sheffield's extremal Brownian Gibbs line ensemble conjecture. As a convex set, the extremal points of $\mathscr{G}_{\Theta}$ are exactly $\{\mathcal{L}^c\mid c\in\mathbb{R}\}$.

This conjecture would classify stationary Brownian Gibbs line ensembles after the parabolic shift and could provide an invariance principle for convergence to the Airy line ensemble, with applications to KPZ universality. The cited work proves that the displayed $\mathcal{L}^c$ are extremal, reducing the conjecture to uniqueness at each prescribed mean height; the full characterization remains open in the supplied text.