Optimal regularity conjecture for two-dimensional KPZ scaling limits

For any $\delta>0$, let $\mathcal{C}_{\mathfrak{s};\mathrm{loc}}^{-\delta}(\mathbf{R}_{>0}\times\mathbf{T}^{2})$ and $\mathcal{C}^{-\delta}(\mathbf{T}^{2})$ denote the corresponding local parabolic and spatial negative Hölder spaces. Optimal regularity conjecture. The tightness statements in the paper's quantitative theorem hold in the spaces

$$\mathcal{C}_{\mathfrak{s};\mathrm{loc}}^{-\delta}(\mathbf{R}_{>0}\times\mathbf{T}^{2})$$

and

$$\mathcal{C}^{-\delta}(\mathbf{T}^{2}),$$

respectively, for every $\delta>0$. The theorem currently proves tightness only in the weaker spaces with regularities $-2-\delta$ and $-1-\delta$; improving these bounds to arbitrarily small negative regularity remains open.