Balla's conjecture for reciprocal odd integers

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

Balla's conjecture for $1/\alpha$ is odd. For every integer $k\geq 1$ and every $d\geq 1$,

$$N_{\frac{1}{2k+1}}(d)\leq \max\left\{\binom{4k^2+4k}{2},\left\lfloor\frac{(k+1)(d-1)}{k}\right\rfloor\right\}.$$

The paper presents this as the natural remaining question after proving Balla's general conjecture in two cases and disproving it for infinitely many values of $\alpha$. Its status is not resolved in the supplied text.