Odd-length diagonal-twist spectral 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 eigenvalue degeneracies. Odd-length diagonal-twist conjecture. For odd $N$ and $0<|x|<\sqrt{2}$, the Hamiltonian has a single negative, doubly degenerate ground-state eigenvalue; all other eigenvalues are non-negative, and the first excited state is the non-degenerate eigenvalue $E=0$. For $x=0$ or $|x|\geq\sqrt{2}$, the spectrum is non-negative and $E=0$ is the ground-state eigenvalue, with degeneracy $2N+1$ for $x=0$, three for $|x|=\sqrt{2}$, and one for $|x|>\sqrt{2}$. The claim is based on computed spectra and remains conjectural in the supplied text.