Main asymptotic conjecture for spherical two-distance sets

Fix $-1\le\beta<0\le\alpha<1$. For a spherical two-distance set in $\mathbb R^d$ with inner products in $\{\alpha,\beta\}$, let $N_{\alpha,\beta}(d)$ be the maximum possible size. For $\lambda=(1-\alpha)/(\alpha-\beta)$ and $p=\lfloor-\alpha/\beta\rfloor+1$, let $k_p(\lambda)$ denote the relevant signed-graph eigenvalue multiplicity parameter. Main conjecture.

$$\lim_{d\to\infty}\frac{N_{\alpha,\beta}(d)}{d}=\begin{cases}\displaystyle\frac{k_p(\lambda)}{k_p(\lambda)-1},&\text{if }k_p(\lambda)<\infty,\\1,&\text{otherwise}.\end{cases}$$

The conjecture is presented as the sharp form of the preceding lower bounds. The paper states that all cases are proved in its main theorem, but the supplied text does not identify the precise cases or establish the full conjecture, so its overall resolution should be checked against that theorem.