The odd periodic constant-term conjecture for the loop model

Let $F_{2n+1}(x)$ be the generating polynomial for the odd-sized periodic ground state, and let ${\rm AHT}_{m}$ denote the corresponding alternating-sign-triangle sum rule. Odd periodic constant-term conjecture. The conjectured value at the constant term is

$$F_{2n+1}(0)=\frac{{\rm AHT}_{2n}}{{\rm AHT}_{2n+1}}.$$

The claim concerns the normalized sum of components whose link patterns produce no loops when paired with the distinguished link pattern. The source notes that it would be equivalent to a conjecture for the punctured even periodic model if the relevant component sum could be identified with that model's sum rule.