Ilman's rigidity conjecture for cylindrical self-shrinking ends

Let $\Sigma\subset\mathbb R^3$ be a smooth complete embedded self-shrinker with at most quadratic area growth. An end is asymptotic to a cylinder if it satisfies the cylindrical alternative: for some vector $v$, as $\tau\longrightarrow +\infty$, the translated end $U-\tau v$ converges to

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

Ilman's cylindrical rigidity conjecture. If one end of $\Sigma$ is asymptotic to a cylinder, then $\Sigma$ is isometric to the self-shrinking cylinder.

This is a rigidity statement intended to classify complete embedded self-shrinkers possessing a cylindrical end; the source presents it as a conjecture, with no resolution given.