Sufficiency of R(A)=1 for balance or antibalance

Establish whether the equality of the ratio R(A) = K(e^{-A}) / K(e^{-|A|}) to 1, computed in the Schatten p-norm with p=1 for the exponential matrices of the signed adjacency matrix A and its entrywise absolute value |A|, implies that the underlying signed graph G is either structurally balanced or antibalanced. Concretely, prove the sufficiency direction that if R(A)=1 then G is balanced or antibalanced.

Background

The paper defines the global balance index κ(G) of a signed graph G via κ(G) = tr[e^A] / tr[e^{|A|}], where A is the signed adjacency matrix. It also introduces the ratio R(A) between the condition numbers (in trace norm) of e^{-A} and e^{-|A|}, and shows R(A) = κ(G)·κ(-G)/κ(-|G|) (Proposition 2).

Proposition 3 establishes a necessary condition: if G is balanced or antibalanced, then R(A)=1. The authors then propose the converse direction as a conjecture, supported by numerical checks on complete signed networks of various sizes, but without a formal proof.

Resolving this sufficiency would provide a sharp characterization linking a numerical conditioning measure to structural balance properties of signed networks, strengthening the conceptual bridge between spectral conditioning and signed-graph topology.