Half-turn-symmetric ground-state conjecture for odd periodic dense O(1) loops

Let $N=2n+1$ be odd and impose periodic boundary conditions. Let $\Pi^{\mathrm{HT}}_{2n+1}$ be the set of half-turn-symmetric pairing patterns, identified with the basis of the dense $O(1)$ loop model, and let $A^{\mathrm{HT}}_{2n+1}(\pi)$ denote the number of half-turn-symmetric alternating-sign-matrix configurations with pairing pattern $\pi$.

Half-turn-symmetric ground-state conjecture. The vector

$$\boldsymbol{\Psi}=\sum_{\pi\in\Pi^{\mathrm{HT}}_{2n+1}} A^{\mathrm{HT}}_{2n+1}(\pi)\,\pi$$

is the ground state vector of the periodic dense $O(1)$ loop model.

This is the paper's first new conjecture and extends the ground-state/combinatorial correspondence to odd periodic size with half-turn symmetry. No resolution is supplied in the source.