Jiang–Polyanskii conjecture for infinite spectral radius order

Fix $\alpha\in(0,1)$, let $N_\alpha(d)$ be the maximum number of equiangular lines in $\mathbb{R}^d$ with common angle parameter $\alpha$, and set

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

For a real $\lambda>0$, let $k(\lambda)$ be the smallest number of vertices of a graph whose spectral radius is exactly $\lambda$, with $k(\lambda)=\infty$ if no such graph exists.

Jiang–Polyanskii conjecture. If $k(\lambda)=\infty$, then

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

The paper explains that its main theorem gives only $N_\alpha(d)=d+O_\alpha(d/\log\log d)$ in this case, while the conjectured bounded excess over $d$ remains open. It also notes that the conjecture had been verified except when $\lambda$ is a totally real algebraic integer that is largest among its conjugates.