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

Let $B^n$ be the unit ball, $S^{n-1}=\partial B^n$, and let $\nu$ denote the unit outer normal vector on $S^{n-1}$. For parameters $q$ and $\lambda$, consider a positive function $u\in C^{\infty}(\overline{B^n})$ satisfying

$$\left\{\begin{array}{ll} \Delta u=0 &\operatorname{in}\ B^n,\\ u u+\lambda u=u^q &\operatorname{on}\ S^{n-1}. \end{array}\right.$$

Guo–Wang's conjecture. If $1<q<\frac{n}{n-2}$ and $0<\lambda\leq\frac{1}{q-1}$, then $u$ is constant.

This conjecture asks for uniqueness of positive harmonic solutions of a nonlinear boundary-value problem in the unit ball. The paper from which the statement is taken presents a proof of Guo–Wang's conjecture, so the conjecture is solved.