Colding-Minicozzi entropy lower-bound conjecture for closed self-shrinkers

Let $\Sigma\subset \mathbb R^N$ be a closed $n$-dimensional self-shrinker. The Colding-Minicozzi entropy should satisfy

$$\sqrt{2}=\lim_{n\to \infty} \left(\frac{2n}{4\pi e}\right)^{\frac{n}{2}}|\mathbb{S}^n|<\left(\frac{2n}{4\pi e}\right)^{\frac{n}{2}}|\mathbb{S}^n|=\lambda_{CM}^n[\mathbb{S}^n]\leq \lambda_{CM}^n[\Sigma].$$

This conjecture concerns the entropy-minimizing closed self-shrinker and asserts that the round $n$-sphere gives the lower bound. Its resolution is not indicated in the source.