Uniqueness conjecture for closed CAMC hypersurfaces under conditions (I), (II), or (III)

Let $\gamma:S^n\to {\mathbb R}_{>0}$ be a $C^2$ function, and let $X:M\to {\mathbb R}^{n+1}$ be a closed CAMC hypersurface. For each $r=1,\ldots,n$, assume that the $r$-th anisotropic mean curvature of $X$ for $\gamma$ is integrable. Let $\xi_\gamma:S^n\to {\mathbb R}^{n+1}$ denote the Cahn–Hoffman map. The uniqueness conjecture. If $X$ satisfies at least one of conditions (I), (II), or (III) above, then $X(M)$ is a subset of a homothety of $\xi_\gamma(S^n)$. This conjecture concerns uniqueness of closed CAMC hypersurfaces under the stated conditions; the conditions are not reproduced in the supplied excerpt, so their precise hypotheses should be checked against the paper.