Asymptotic structure conjecture for two-dimensional self-shrinkers

Let $\Sigma\subset\mathbb{R}^3$ be a complete noncompact properly embedded self-shrinker with finite genus. Here a self-shrinker satisfies

$$H=\frac{1}{2}\langle\mathbf{x},\mathbf{n}\rangle,$$

where $\mathbf{n}$ is the unit normal, $H=\operatorname{div}\mathbf{n}$ is the mean curvature, and $\mathbf{x}$ is the position vector. Suppose that outside some closed ball, $\Sigma$ decomposes into finitely many connected components $U_i$.

Asymptotic structure conjecture. For each $U_i$, either, as $\lambda\to\infty$, $\lambda^{-1}U_i$ converges locally smoothly to a cone smooth except at the origin $\mathbf{0}$, or there exists a unit vector $\mathbf{v}_i$ such that, as $\lambda\to\infty$, $U_i-\lambda\mathbf{v}_i$ converges locally smoothly to the self-shrinking cylinder with axis parallel to $\mathbf{v}_i$.

This conjecture concerns the classification of ends of properly embedded self-shrinkers and their geometry at infinity. It is motivated by rigidity results for self-shrinkers asymptotic to the same cone or cylinder, but the source provides no resolution, so the conjecture remains open.