Wang's type II Yamabe rigidity conjecture

Let $(M^n,g)$ be a smooth compact Riemannian manifold with $Ric\geq 0$ and second fundamental form $\Pi\geq 1$ on its boundary $\Sigma$. Let $u\in C^{\infty}(M)$ be a positive solution of

$$\begin{array}{ccc} \Delta u=0 & \text{on} & M,\\ \frac{\partial u}{\partial \nu}+\lambda u=u^q & \text{on} & \Sigma. \end{array}$$

Here $\lambda>0$ and $1<q\leq \frac{n}{n-2}$. Wang's type II Yamabe rigidity conjecture. If $\lambda\leq \frac{1}{q-1}$, then $u$ must be constant unless $q=\frac{n}{n-2}$, $M$ is isometric to the closed unit ball $\overline{\mathbb B^n}\subset\mathbb R^n$, and $u$ corresponds to

$$u_a(x)=\left[\frac{2}{n-2}\frac{1-|a|^2}{1+|a|^2|x|^2-2x\cdot a}\right]^{(n-2)/2}$$

for some $a\in\mathbb B^n$. This is the proposed sharp rigidity statement for the type II Yamabe problem under nonnegative Ricci curvature and strictly convex boundary; the supplied text does not establish its resolution.