Norm formula conjecture for odd XXZ ground states

Let $N=2n+1$ be odd, let ${\mathcal N}_n$ denote the norm of the ground state, and normalize the state by $\psi_{0\ldots 01\ldots 1}=1$. Let

$$A_n=\prod_{j=0}^{n-1}\frac{(3j+1)!}{(n+j)!}.$$

Norm formula conjecture. The norm is

$${\mathcal N}_n=\frac{\sqrt{3^n}}{2^n}\,\frac{2\cdot5\cdots(3n-1)}{1\cdot3\cdots(2n-1)}\,A_n.$$

The formula agrees with the computed values for odd $N\leq17$ and is presented as one of the paper's conjectures; no general proof is given here.