Bukh's odd-reciprocal conjecture for equiangular lines

Let $N_\alpha(n)$ denote the maximum number of equiangular lines in $\mathbb{R}^n$ with common angle $\arccos\alpha$. Bukh's conjecture. If $1/\alpha$ is an odd natural number, then

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

The paper states that this is the first part of an earlier conjecture and that it was also raised by Bukh. The result proved in the paper covers the relevant range through the spectral-radius-order formula, so the conjecture is consistent with and follows from that theorem.