Conjecture on positive boundary value solutions under Ricci and convexity bounds

Let $M^n$ be a compact Riemannian manifold with $Ric\geq 0$ and second fundamental form $\Pi\geq 1$ on $\partial M$. Let $u$ be a positive solution of

$$\Delta u=0 \text{ on } M,\qquad \frac{\partial u}{\partial\nu}+\lambda u=u^q \text{ on } \partial M,$$

where $\lambda>0$ and $1<q\leq n/(n-2)$. The positive-solution conjecture. If $0<\lambda\leq 1/(q-1)$, then $u$ must be constant unless $q=n/(n-2)$, $M$ is isometric to $\overline{\mathbb{B}^n}\subset\mathbb{R}^n$, 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 a rigidity conjecture for the critical nonlinear boundary problem; the supplied text does not indicate whether it has been resolved.