The periodic odd-size O(1) loop-model 1-sum conjecture

Consider the dense O(1) loop model with periodic boundary conditions and odd system size $L=2n-1$, and let $S_{2n-1}^{(1)}$ be the sum of the components of its groundstate wavefunction. Define

$$\mathcal N_n=\left(\frac{\sqrt 3}{2}\right)^n \frac{2\cdot5\cdots(3n-1)}{1\cdot3\cdots(2n-1)}A_n,$$

where $A_n$ is the number of $n\times n$ alternating sign matrices. The periodic odd-size 1-sum conjecture.

$$S_{2n-1}^{(1)}=\mathcal N_n^2.$$

This extends the alternating-sign-matrix and plane-partition pattern to odd periodic loop-model sizes. It is presented as a finite-size conjecture in the source and remains open.