Guo–Wang's conjecture for positive harmonic functions on the unit ball

Let $n\geq 3$, let $\mathbb{B}^n\subset\mathbb{R}^n$ be the unit ball with boundary sphere $\mathbb{S}^{n-1}$, and let $u\in C^{\infty}(\mathbb{B}^n)$ be positive and satisfy

$$\begin{cases} \Delta u=0 & \operatorname{in} \mathbb{B}^n,\\\\ \dfrac{\partial u}{\partial\nu}+\lambda u=u^q & \operatorname{on} \mathbb{S}^{n-1}, \end{cases}$$

where $1<q\leq\dfrac{n}{n-2}$ and $0<\lambda\leq\dfrac{1}{q-1}$ are constants. Guo–Wang's conjecture. Either $u$ is constant, or $q=\dfrac{n}{n-2}$, $\lambda=\dfrac{n-2}{2}$, and, for some $a\in\mathbb{B}^n$,

$$u(x)=\left(\dfrac{n-2}{2}\dfrac{1-|a|^2}{|a|^2|x|^2-2\langle x,a\rangle+1}\right)^{\frac{n-2}{2}}.$$

This is the model-space version of Wang's conjecture. The paper gives partial verification, while the conjecture in the full stated parameter range remains open; related results resolve the two-dimensional analogue and the ball case cited in the paper.