Conjecture on concentration for sign-changing radial solutions as beta increases to one

Let $\Lambda^*<\lambda_*<\Lambda_1$ and set $\beta_*=1$. Let $S_{1,\lambda,\beta}$ denote the class of radial sign-changing solutions with the index specified in the paper, and let behavior (v) denote the corresponding concentration and weak-limit alternative from Theorem A1. Concentration conjecture. There exist sequences

$$(\lambda_n)\subset(0,\Lambda_1),\qquad (\beta_n)\subset(0,1),$$

and solutions $u_n$ such that

$$(\lambda_n,\beta_n)\to(\lambda_*,\beta_*),\qquad u_n\in S_{1,\lambda_n,\beta_n}\quad\text{for all }n\in\mathbb N,$$

and $(u_n)$ behaves as in (v) of Theorem A1. This predicts the counterpart, for $\beta_n\uparrow1$, of concentration phenomena already identified in the opposite parameter direction; the supplied text gives no resolution.