Sum-of-components formula conjecture for odd XXZ ground states

Let $N=2n+1$ be odd, let ${\mathcal N}_n$ be the norm of the ground state, and let the state vector have components in the spin-configuration basis. Sum-of-components conjecture. The sum of all components of the state vector is

$$(\sqrt{3})^n{\mathcal N}_n.$$

The formula is inferred from the computed cases in the paper and is not proved there.