Jiang et al.'s asymptotic conjecture for spherical two-distance sets

Let $-1 \leq \beta < 0 \leq \alpha < 1$. A spherical two-distance set in $\mathbb{R}^d$ is a collection of unit vectors whose pairwise inner products lie in $\{\alpha,\beta\}$. Let $N_{\alpha,\beta}(d)$ be the maximum size of such a set, and define

$$\lambda = \frac{1-\alpha}{\alpha-\beta}, \qquad p = \left\lfloor-\frac{\alpha}{\beta}\right\rfloor+1.$$

For $p\in\mathbb{N}^+$ and $\lambda>0$, let $\mathcal{G}_p(\lambda)$ be the relevant family of $p$-colorable signed graphs and define

$$k_p(\lambda)=\inf\left\{\frac{|G|}{\operatorname{mult}(-\lambda,G)}:G\in\mathcal{G}_p(\lambda)\right\},$$

where $\operatorname{mult}(-\lambda,G)$ is the multiplicity of $-\lambda$ as an eigenvalue of $G$. Jiang et al.'s asymptotic conjecture.

$$N_{\alpha,\beta}(d)= \begin{cases} \dfrac{k_p(\lambda)d}{k_p(\lambda)-1}+o(d) & \text{if }k_p(\lambda)<\infty,\\ d+o(d) & \text{otherwise}. \end{cases}$$

This conjecture predicts the asymptotically optimal size of spherical two-distance sets outside the equiangular case. The limit is known in some regimes, including $p\leq 2$ or $\lambda<\lambda^*$, but the general case remains open.