Odd-length XXZ spin-chain ground-state energy conjecture

Let $H$ be the Hamiltonian referred to as ${\rm (H1)}$, with an odd number $N$ of sites, and let $E$ and $S_z$ denote an energy eigenvalue and the total spin in the $z$-direction. Ground-state energy conjecture. The ground states of $H$ for arbitrary odd $N$ have energy

$$E=-\frac{3N}{2}$$

and satisfy $S_z=\pm\frac12$. This conjecture was formulated earlier and supported by numerical calculations; the paper confirms it for the odd sizes examined, but the general assertion is not established here.