Uniqueness of Brownian Gibbs line ensembles with prescribed mean height

Let $\mathscr{G}_{\Theta}$ be the set of Brownian Gibbs line ensemble measures whose parabolically shifted lines are horizontal shift-invariant. For a real number $c$, consider Brownian Gibbs line ensembles $\mathcal{L}\in\mathscr{G}_{\Theta}$ and their top line $\mathcal{L}_1$.

Uniqueness conjecture. For every $c\in\mathbb{R}$, there is a unique Brownian Gibbs line ensemble $\mathcal{L}\in\mathscr{G}_{\Theta}$ such that

$$\mathbb{E}\left[\mathcal{L}_1(x)+2^{-1/2}x^2\right]=c$$

for all $x\in\mathbb{R}$.

This is the reduction of the preceding extremal classification conjecture after extremality of the shifted Airy line ensembles has been proved. It is presented as the remaining uniqueness problem, and no resolution is supplied in the source.