Plane-or-cylinder asymptotic conjecture for self-shrinkers

Let $\Sigma\subset\mathbb{R}^3$ be a complete noncompact properly embedded self-shrinker with finite genus, and define its Gaussian surface area by

$$\mathbf{F}[\Sigma]=(4\pi)^{-1}\int_\Sigma\operatorname{e}^{-\lvert\mathbf{x}\rvert^2/4}\,d\mathcal{H}^2.$$

Assume that $\mathbf{v}_0\in\mathbb{R}^3\setminus\{\mathbf{0}\}$ satisfies $\mathbf{F}[\Sigma-\lambda\mathbf{v}_0]\geq 1$ for all $\lambda>0$.

Plane-or-cylinder asymptotic conjecture. As $\lambda\to\infty$, $\Sigma-\lambda\mathbf{v}_0$ converges locally smoothly to a plane or a self-shrinking cylinder of multiplicity one.

This is a reformulation intended to address the formation of singularities of self-shrinking mean-curvature flows at time $0$. The source gives no evidence that the statement has been proved or disproved, so it remains open.