Twisted XXZ component-ratio conjecture

Let $|\Psi^{(2n)}\rangle$ be the normalized ground-state vector of the twisted XXZ chain with $2n$ sites, and let $A_n$ denote the number of alternating-sign $n\times n$ matrices. The components with smallest and largest absolute values are respectively $\Psi^{(2n)}_{\downarrow\ldots\downarrow\uparrow\ldots\uparrow}$ and $\Psi^{(2n)}_{\downarrow\uparrow\ldots\downarrow\uparrow}$. The twisted XXZ component-ratio conjecture. Their absolute-value ratio is

$$\left|\frac{\Psi^{(2n)}_{\downarrow\uparrow\ldots\downarrow\uparrow}}{\Psi^{(2n)}_{\downarrow\ldots\downarrow\uparrow\ldots\uparrow}}\right| =\left(\frac{2}{\sqrt{3}}\right)^{n-1} \frac{1\cdot3\cdot5\cdots(2n-3)(2n-1)}{1\cdot4\cdot7\cdots(3n-5)(3n-2)}A_n.$$

This conjecture gives a closed combinatorial formula for an extremal component ratio; the source reports only verification for the available data.