Balla's conjecture on equiangular lines

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

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

Let $\kappa_{\lambda}$ denote the spectral radius order of $\lambda$, and assume that $\kappa_{\lambda}<\infty$.

Balla's conjecture. For any $d\geq 1$,

$$N_{\alpha}(d)\leq \max\left\{\frac{(1-\alpha^2)(1-2\alpha^2)}{2\alpha^4},\left\lfloor\frac{\kappa_{\lambda}(d-1)}{\kappa_{\lambda}-1}\right\rfloor\right\}.$$

The conjecture was previously verified for $\alpha\in\left\{\frac{1}{3},\frac{1}{5},\frac{1}{1+2\sqrt{2}}\right\}$, and this paper proves it for $\alpha=\frac{1}{1+2\sqrt{3}}$ and $\alpha=\sqrt{5}-2$. However, it fails for infinitely many values of $\alpha$, so the general conjecture is refuted.