Jiang–Tidor–Yao–Zhang–Zhao bounded-error conjecture for equiangular lines

Let $\alpha\in(0,1)$, define

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

and let $\kappa=\kappa(\lambda)$ be the spectral radius order of $\lambda$, namely the least number of vertices of a graph whose largest adjacency-matrix eigenvalue is $\lambda$, or $\infty$ if no such graph exists. Jiang–Tidor–Yao–Zhang–Zhao conjecture. If $\kappa=\infty$, then

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

The source states that this remains open to determine the set of $\lambda$ for which the conjecture holds, although it has been confirmed when $\lambda$ is not a totally real algebraic integer.