The AKPZ universality-class conjecture for two-dimensional growth models

Let $v(\cdot)$ be the speed function of a reasonable $(2+1)$-dimensional growth model, and let $H_\rho$ be the Hessian of $v(\cdot)$ computed at the slope $\rho$. Write $\alpha$ for the spatial growth exponent and $\beta$ for the temporal growth exponent. AKPZ universality conjecture. If $\det(H_\rho)>0$, then height fluctuations grow in time as $t^\beta$ for some model-independent $\beta>0$, and height fluctuations in stationary states grow as distance to the power $2\beta/(\beta+1)$. If instead $\det(H_\rho)\leq 0$, then $\beta=\alpha=0$ and stationary states have the same spatial height correlations as a massless Gaussian field. This conjecture predicts a distinction between the isotropic KPZ and anisotropic KPZ universality classes: the former has nonzero growth exponents, while the latter has logarithmic fluctuations and Gaussian-field spatial correlations. The assertion remains unproved for the relevant $(2+1)$-dimensional growth models.