Even-length diagonal-twist ground-state conjecture

Let $N$ be the length of the spin-one chain and let $x$ be its anisotropy parameter. Consider the Hamiltonian with diagonal twist and its spectrum, including eigenvalue degeneracies. Even-length diagonal-twist conjecture. For even $N$ and every $x$, the spectrum is non-negative, with ground-state eigenvalue $E=0$. This eigenvalue is non-degenerate when $x eq 0$ and has degeneracy $N+1$ when $x=0$. The conjecture gives a proposed complete description of the ground-state energy and degeneracy for even chains; its resolution is not stated in the supplied text.