The odd periodic special-value conjecture for the loop model

Let $F_{2n+1}(x)$ be the generating polynomial associated with the odd-sized periodic ground state, and let ${\rm AV}_{2n+1}$ and ${\rm AHT}_{2n+1}$ denote the corresponding alternating-vertex and alternating-sign-triangle enumerations. Odd periodic special-value conjecture. The conjectured special values are

$$F_{2n+1}(-1)=\frac{{\rm AV}_{2n+1}}{{\rm AHT}_{2n+1}},\qquad F_{2n+1}(2)=\frac{2^{2n}{\rm AV}_{2n+1}}{{\rm AHT}_{2n+1}}.$$

The source reports these as unexplained observations for the odd periodic case, so their general validity remained open there.