Cylindrical self-shrinker rigidity conjecture

Let $\Sigma^n\subset\mathbb{R}^{n+1}$ be a smooth embedded self-shrinker with entropy at most $\alpha$, where $\alpha>1$. Let $B_R$ be the Euclidean ball of radius $R$, and let $H$ denote the mean curvature. Cylindrical self-shrinker rigidity conjecture. For every $n$ and $\alpha>1$, there exists $R=R(n,\alpha)>0$ such that if $H\geq 0$ on $B_R\cap\Sigma$, then $H\geq 0$ on all of $\Sigma$ and consequently $\Sigma$ is a generalized cylinder $\mathbb{S}^k\times\mathbb{R}^{n-k}$. The claim is part of work described as in progress; the source does not report a resolution.