The no cylinder conjecture for embedded shrinking solitons

Let $\Sigma$ be an embedded shrinking soliton in $\mathbb{R}^3$, and suppose that $\Sigma$ is not a round cylinder. No cylinder conjecture. Then $\Sigma$ cannot have an end asymptotic to a cylinder.

This conjecture rules out cylindrical ends for noncylindrical embedded shrinking solitons in $\mathbb{R}^3$. Together with Ilmanen’s multiplicity one conjecture, it is used in the paper to derive bounded intrinsic diameter for two-dimensional mean curvature flows; the conjecture remains unresolved in the supplied text.