Existence of representative zero-energy states

Let $H_N(x,y)$ be the Hamiltonian on $N$ sites, with twist angle $\phi$, and let $S_N^3$ denote the total third spin component. The representative singlet conjecture. For every $N$ and twist angle $\phi=\pi$, there exist parameters $x,y$ with $x\ne y$ for which the Hamiltonian has exactly one zero-energy state in the sector $S_N^3=0\bmod 2$. For twist angle $\phi=0$, there exist parameters $x,y$ with $x\ne y$ for which the Hamiltonian has no zero-energy states. If true, the symmetry and conjugation arguments imply the corresponding statements throughout the non-light-like region $|x|\ne|y|$.