Asymptotic linear bound conjecture for the second Lasserre level

Let $N_\alpha(n)$ denote the maximum number of equiangular lines in $\mathbb{R}^n$ with common angle $\arccos\alpha$, and let $\mathrm{las}_2(n)$ be the optimal objective of the second level of the Lasserre hierarchy for bounding $N_\alpha(n)$. Let $\alpha\in(0,1)$.

Asymptotic linear bound conjecture. The optimal objective satisfies

$$\mathrm{las}_2(n)\leq c_\alpha+\frac{1+\alpha}{2\alpha}n$$

for $n\geq n_\alpha$, where $c_\alpha$ does not depend on the dimension.

This conjecture predicts an asymptotically linear second-level semidefinite-programming bound with the same slope as the currently known general bound. The paper proves it for $\alpha=1/a$ with $a=3,5,7,9,11$, while the assertion for general $\alpha\in(0,1)$ remains open.