The stable-region splitting conjecture for higher-dimensional hyperplanes

Let $\Sigma=\mathbb{R}^n\subset\mathbb{R}^{n+1}$ be a self-shrinker with $n\geq 3$. Let $r_1$ be the dimension-dependent radius from the cited remark. Then there should exist a number $r_0=r_0(n)$ with $r_0<r_1$ such that, for every $r\in(r_0,r_1)$, the $(n-1)$-sphere of radius $r$ centered at the origin splits $\Sigma$ into two stable regions. Stable-region splitting conjecture. There exists $r_0=r_0(n)<r_1$ such that every such sphere with radius in $(r_0,r_1)$ splits the hyperplane into two stable regions. The analogous statement is known for the plane in $\mathbb{R}^3$, but the source says that the argument has not been reproduced in higher dimensions.