Asymptotic structure of the minimizer near a singular point

Let $H:\mathbb{R}^2\to\mathbb{R}$ be the scaling-limit function associated with the minimizer near a singular point of the minimal height function. The asymptotic structure conjecture. The function $H$ is once differentiable. There is a smooth curve separating $\mathbb{R}^2$ into regions where $H$ is linear and regions where $H$ is smooth with positive definite matrix of second derivatives. This describes the expected local separation between frozen and liquid regions near a singular point; the source does not state a resolution.