Anti-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 anti-diagonal twist. Anti-diagonal-twist conjecture. For every $N$ and every $x$, its spectrum is non-negative and $E=0$ is the ground-state eigenvalue. This eigenvalue is non-degenerate for $x\neq 0$ and has degeneracy three for $x=0$. The conjecture is presented as one of three claims supported by finite-size spectral computations; its resolution is not stated in the supplied text.