The spectral-radius-order conjecture for equiangular lines

Let $N_\alpha(n)$ denote the maximum number of equiangular lines in $\mathbb{R}^n$ with angle $\arccos\alpha$. For $\alpha\in(0,1)$, define

$$\lambda=\frac{1-\alpha}{2\alpha}$$

and let $k(\lambda)$ be the smallest order of a graph with spectral radius $\lambda$, setting $k(\lambda)=\infty$ if no graph has spectral radius $\lambda$. Spectral-radius-order conjecture.

$$N_\alpha(n)=\frac{k(\lambda)}{k(\lambda)-1}\,n+O(1).$$

If $k(\lambda)=\infty$, this means $N_\alpha(n)=n+O(1)$. The conjecture is presented as a stronger form of the odd-reciprocal conjecture, while the paper proves it when $\lambda\le\sqrt{2+\sqrt{5}}$ and gives an exact bound in one further algebraic-integer case; it remains open in general.