Find dense subgraphs of the theta-free octagon construction without parity obstructions

Construct, or determine the existence of, sufficiently dense subgraphs of the Verstraëte–Williford graph $\mathcal W_q$ that admit a signing in which every $8$-cycle is unbalanced, despite the fact that $\mathcal W_q$ itself has an odd dependence among its $8$-cycles.

Background

The Verstraëte–Williford graphs $\mathcal W_q$ are $\Theta_{4,3}$-free and have the conjectural octagon density order $q^5$ edges on $q^4$ vertices, so they satisfy the necessary theta-freeness condition for a balanced-octagon-free signing. The paper proves that, for an infinite family of parameters, the full graph contains a subdivision of $K_4$ producing an odd dependence among its $8$-cycles; consequently, no signing can make every $8$-cycle unbalanced. The authors leave open whether sufficiently dense subgraphs can avoid this parity obstruction.