Signature bound by cycle counts

Prove that every graph G with signature s(G)=n^+(G)-n^-(G) satisfies -c_3(G) ≤ s(G) ≤ c_5(G), where c_3(G) counts cycles whose lengths are congruent to 3 modulo 4 and c_5(G) counts cycles whose lengths are congruent to 1 modulo 4.

Background

The paper records an unresolved conjecture attributed to Ma, Yang, and Li concerning the relationship between graph signature and counts of cycles in two residue classes modulo 4. It is presented as a related inertia question, distinct from the paper’s principal conjecture.