Odd-cycle characterization of bases for analytic spread Jacobians

Let $G$ and $H$ be graphs on $n$ and $m$ vertices, respectively, with $n,m\geq 3$. Let $J_{G,H}$ be the associated binomial edge ideal, let $\ell(J_{G,H})$ denote its analytic spread, and let $\mathcal{H}(G,H)$ be its Jacobian. Odd-cycle basis conjecture. The rows of $\mathcal{H}(G,H)$ indexed by $E(G)\times E(H)$ form a basis for the row space of $\mathcal{H}(K_n,K_m)$, equivalently

$$\ell(J_{G,H})=|E(G)||E(H)|=|V(G)||V(H)|,$$

if and only if both $G$ and $H$ are disjoint unions of odd cycles. This proposes a combinatorial characterization of when the indicated Jacobian rows form a basis, but the source supplies computational motivation rather than a proof or resolution.