The critical two-dimensional KPZ scaling-limit conjecture

Let $\mathscr{Z}^\vartheta_t(1,B_\varepsilon(x))$ denote the mass of the critical $2$-dimensional stochastic heat flow at time $t$ in the ball $B_\varepsilon(x)$, and let $C_c(\mathbb{R}^2)$ be the space of compactly supported continuous functions. For a test function $\psi\in C_c(\mathbb{R}^2)$, consider the centered, rescaled spatial integral

$$\int_{\mathbb{R}^2}\frac{1}{b(\varepsilon)}\left(\log \mathscr{Z}^\vartheta_t(1,B_\varepsilon(x))-E[\log \mathscr{Z}^\vartheta_t(1,B_\varepsilon(x))]\right)\psi(x)\,\mathrm{d}x.$$

Critical KPZ scaling-limit conjecture. There exists a non-random function $b(\varepsilon)>0$ with $b(\varepsilon)\to\infty$ as $\varepsilon\to0$ such that, for every $\psi\in C_c(\mathbb{R}^2)$, the displayed random variable converges weakly to a nontrivial random variable $\mathfrak{h}_t(\psi)$.

This formulates the expected nontrivial scaling limit for spatial averages of the logarithmic mass in the critical two-dimensional KPZ equation. The paper establishes integrability and upper bounds for the lower tail of the logarithmic mass, but does not identify such a normalization or prove the asserted convergence.