Quadratic generation conjecture for toric ideals of twinned order polytopes

Let $P$ and $Q$ be finite posets with $|P|=|Q|=d$. For the toric ideal $I_{\Omega(P,-Q)}$ associated with the twinned order polytope $\Omega(P,-Q)$, quadratic generation conjecture. the ideal $I_{\Omega(P,-Q)}$ is generated by quadratic binomials. The paper notes that this holds in the examples considered, although the corresponding quadratic Gröbner-basis property can fail.