Jiang–Polyanskii conjecture on the bounded secondary term for equiangular lines

For $\alpha\in(0,1)$, let $\lambda=(1-\alpha)/(2\alpha)$, and let $k(\lambda)$ denote the minimum number of vertices of a graph whose largest eigenvalue is exactly $\lambda$, or $\infty$ if no such graph exists. Jiang–Polyanskii conjecture. If $k(\lambda)=\infty$, then

$$N_\alpha(n)=n+O_\alpha(1).$$

The quantity $N_\alpha(n)$ concerns the maximum size of a family of equiangular lines with common angle parameter $\alpha$. The conjecture would determine the secondary term in the $k(\lambda)=\infty$ case, strengthening the known estimate $N_\alpha(n)=n+o(n)$; its status is not resolved in the supplied source.