The Airy sheet extension conjecture

Let $\mathcal{A}$ be the Airy line ensemble, and define $\mathcal{H}:\mathbb{R}^2\to\mathbb{R}$ deterministically from $\mathcal{A}$ by the formulas in the source on $(0,\infty)\times\mathbb{R}$, by $\mathcal{H}(0,y)=\mathcal{A}_1(y)$, and on the negative half-plane by applying the reflected formulas. Airy sheet conjecture. The function $\mathcal{H}$ is an Airy sheet. This would identify the whole Airy sheet as a deterministic functional of the Airy line ensemble; the source indicates that the relevant problem is last passage through the other tableau of RSK, which appears harder to analyze than the Brownian melon.