The KPZ sheet-to-Airy-sheet conjecture

Let $\mathcal{Z}(s,x;t,y)$ denote the continuum directed random polymer partition function, and define

$$\mathfrak{h}_{t}(x,y):=t^{1/3}\left[\log\mathcal{Z}(t^{2/3}x,0;t^{2/3}y,t)+\frac{t}{24}\right].$$

Let $\mathcal{S}(x,y)$ be the parabolic Airy sheet. KPZ sheet-to-Airy-sheet conjecture. As $t\to\infty$,

$$2^{1/3}\mathfrak{h}_{t}(2^{1/3}x,2^{1/3}y)\stackrel{d}{\longrightarrow}\mathcal{S}(x,y)$$

in the uniform-on-compact topology, viewed as functions of $(x,y)$. This conjecture would upgrade the pointwise weak convergence of rescaled continuum directed random polymer paths to process-level convergence; the limiting parabolic Airy sheet is a central universal object in KPZ theory. Its resolution is not supplied in the source.