Sharp asymptotic for dense random circulant graphs

Prove or refute that the expected Lovász number of a dense random circulant graph satisfies [?].

Background

The authors formulate a conjecture based on numerical observations that the expected Lovász number of a dense random circulant graph is asymptotic to [?]. Establishing this would improve the paper's upper bound from [?] to the conjectured sharp scale [?]. The discussion explains that the current restricted-isometry-based proof strategy cannot establish the conjecture because sparse vectors exist in the relevant random Fourier kernel.

References

Based on numerical observations, we formulate the following conjecture.

\begin{conjecture} \label{conj:sharp} Let G be a dense random circulant graph. Then, \begin{equation} \E (G) = (1 + o(1)) \sqrt{n}. \end{equation} \end{conjecture}

— The Lovász number of random circulant graphs  (2502.16227 - Bandeira et al., 22 Feb 2025) in Section 4, Discussion, Conjecture 1