Ramanujan eigenvalue bound for CD(k,q)

Prove that the second-largest eigenvalue of CD(k,q) is at most 2√q for all pairs (k,q).

Background

The paper notes that CD(k,q) is Ramanujan in several cases and that the conjectured bound has been proved for k=4 and for k=5 with odd q. The bound remains unresolved in general.

References

Conjecture 9. (Ustimenko) For all (k, q), CD(k, q) has second largest eigenvalue less than or equal to 2√q.

— Some families of graphs, hypergraphs and digraphs defined by systems of equations  (2503.07915 - Lazebnik et al., 10 Mar 2025) in Conjecture 9, Section 6.6