Li–Nirenberg symmetry conjecture under the Main Assumption and Condition S

Let $M$ be a smooth compact connected embedded hypersurface in $\mathbb R^{n+1}$. The Main Assumption is that whenever $(X',A),(X',B)\in M$ satisfy $A\le B$ and the segment $\{(X',\theta A+(1-\theta)B):0\le\theta\le1\}$ lies in $\overline G$, one has

$$H(X',B)\le H(X',A).$$

Here $H$ is the mean curvature and $G$ is the bounded open set enclosed by $M$. Condition S means that $M$ stays on one side of every hyperplane parallel to the $X_{n+1}$ axis tangent to $M$. Li–Nirenberg's conjecture. If $M$ satisfies the Main Assumption and Condition S, then $M$ is symmetric about a hyperplane $X_{n+1}=c$. The source states that the Main Assumption alone admits counterexamples, while the conjecture is presented without a stated resolution.