Signature-independence conjecture for Adinkra invariant factors

Let an Adinkra be a signed graph with signed Laplacian $L$, and let its invariant factors mean the diagonal entries in the Smith normal form of $L$. A signature is the choice of signs on the edges, and a totally odd signature is one for which every relevant 2-color cycle has odd sign product. Signature-independence conjecture. The invariant factors do not depend on the signature of the Adinkra. The claim is supported by the cases examined in the paper, but no general proof is given.