Vortex-free ground-state conjecture for ladder Hamiltonians

Let $H$ be the Hamiltonian in the paper's equation (Hamiltonian), and let $\widetilde{H}$ be its extended Hamiltonian in equation (HamiltonianExtended). A ladder may be closed or open, and its coupling constants are denoted by $J_{(ij)}$. Vortex-free ground-state conjecture. For a closed ladder, the ground-state energies of $H$ and $\widetilde{H}$ coincide. For a closed or open ladder with the coupling constants $J_{(ij)}$ all positive or all negative, the ground states of $H$ and $\widetilde{H}$ are vortex-free. The conjecture is based on numerical and perturbative calculations for ladder Hamiltonians; the supplied source does not state a resolution.