The classification conjecture for complete self-shrinkers with constant squared second fundamental form

Let $X:M^n\rightarrow\mathbb{R}^{n+1}$ be an $n$-dimensional complete self-shrinker, meaning that its mean curvature vector satisfies

$$\overrightarrow{H}+X^{\perp}=0.$$

Suppose that the squared norm of its second fundamental form is constant. Self-shrinker classification conjecture. $X$ is isometric to one of $\mathbb{R}^n$, $S^k(\sqrt{k})\times\mathbb{R}^{n-k}$ for $0\leq k\leq n-1$, or $S^n(\sqrt{n})$. The conjecture is known in dimension $2$, and in dimension $3$ under additional assumptions that $f_3$ or $f_4$ is constant, but remains difficult in higher dimensions.