Uniqueness conjecture for subcritical nonlinear boundary problems on the ball

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

$$\Delta u=0\quad\text{on }\mathbb{B}^{n},\qquad \frac{\partial u}{\partial\nu}+au=u^{q}\quad\text{on }\mathbb{S}^{n-1}.$$

Uniqueness conjecture. If $1<q<n/(n-2)$ and $0<a\leq 1/(q-1)$, then $u$ is constant.

If true, this uniqueness result would yield the sharp subcritical inequality discussed in the paper; taking $q$ up to the critical exponent would also give the critical case. The cited general formulation on manifolds with nonnegative Ricci curvature and sufficiently convex boundary is recorded separately below.