Radiality of solutions on the complex ball

Let $\mathcal{B}\subset\mathbb{C}^{n}$ be the ball, and consider the equation referred to in the source as equation, with parameter $\gamma$. Let $u_{\gamma}$ denote the radial solution described earlier in the paper. Radiality conjecture for ball solutions. Any solution of this equation on the ball is radial and hence equals the previously described solution $u_{\gamma}$.

For radial solutions, the source states that $u_{\gamma}$ is uniquely determined in the subcritical range $\gamma<n+1$. The conjecture would extend this radial classification to arbitrary solutions in general complex dimension; in dimension one, radiality follows from moving-plane methods.