4 problems

Airy-type characterization conjecture. Then $\mathcal{L}$ is equal in distribution to the $\mathrm{KPZ}_t$ line ensemble, up to an independent affine shift.

Parabolic rigidity conjecture. As $n\to\infty$, the sequence of functions

Asymptotic gap conjecture. Almost surely as $n\to\infty$,

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…