Ilman's asymptotic-end conjecture for embedded self-shrinkers

Let $\Sigma\subset\mathbb R^3$ be a smooth complete embedded self-shrinker with at most quadratic area growth. An end is a connected component of $\Sigma\setminus B_R$ for sufficiently large $R$. For a vector $v_j$, the associated self-shrinking cylinder is

$$\left\{x: \operatorname{dist}(x,\operatorname{span}(v_j))=\sqrt{2}\right\}.$$

Ilman's asymptotic-end conjecture. There exists $R>0$ such that $\Sigma\setminus B_R$ decomposes into a finite number of ends $U_j$, and for each $j$ either:

• as $\lambda\longrightarrow 0+$, $\lambda U_j$ converges locally smoothly to a cone $C_j$ such that $C_j\setminus\{O\}$ is smooth; or
• there is a vector $v_j$ such that, as $\tau\longrightarrow +\infty$, $U_j-\tau v_j$ converges to the cylinder

$$\left\{x:\operatorname{dist}(x,\operatorname{span}(v_j))=\sqrt{2}\right\}.$$

All known examples and numerical evidence support this proposed classification of ends, but the conjecture remains open.