Additional zero-energy states at the elliptic supersymmetric point

Let $H_N$ be the Fateev–Zamolodchikov spin-chain Hamiltonian, with twist angle $\phi$ and total third spin component $S_N^3$. The elliptic zero-mode conjecture. For odd $N=2n+1$, twist angle $\phi=\pi$, and crossing parameter $\eta=\pi/4$, the Hamiltonian possesses two additional zero-energy states in the sector $S_N^3=1\bmod 2$. In the trigonometric limit, these states occur in the sectors $S_N^3=\pm1$. The claim is based on exact diagonalisation for $N=3,5,7$.