Uniqueness conjecture for the multi-layer limiting configuration

Let $(\beta_1,\ldots,\beta_{k-1})$ be the configuration appearing in the $k$-layer limit problem of Theorem defining the corresponding multi-layer radial solution. Uniqueness conjecture. The configuration $(\beta_1,\ldots,\beta_{k-1})$ is unique. The authors note that this is natural because the 1-layer solution of the limit problem has already been proved unique; uniqueness for the general $k$-layer configuration remains conjectural here.