Complete self-shrinkers with constant norm of the second fundamental form in ${\text{RR}}^4$

Let $X:M^3\to {\text{RR}}^4$ be a complete self-shrinker with constant squared norm $S$ of the second fundamental form. We prove that $S=0$ or $S=1$, and that $X(M)$ is isometric to a hyperplane or a standard product ${\text{SSS}}^k(\sqrt{k})\times {\text{RR}}^{3-k}$, $1\le k\le 3$. This proves a conjecture proposed by Cheng and Wei for three-dimensional hypersurfaces.