Rigidity of Immersed Gauss Curvature Shrinkers in Euclidean Space

For every $m\geq2$, we prove that any connected closed oriented smooth immersion $X:M^m\to\mathbb{R}^{m+1}$ satisfying \[ K=\left\langle X,ν\right\rangle \] is an embedding onto the unit sphere. No convexity is assumed. At a point maximizing $|X|$, we find a strictly convex component of $\{K>0\}$.