The open O(1) loop-model 1-sum conjecture

Consider the dense O(1) loop model with open boundary conditions and system size $L$, and let $S_L^{(1)}$ be the sum of the components of its groundstate wavefunction, normalised so that its smallest element is $1$. Let $A_{\rm v}(2n+1)$ and $N_8(2n)$ have the enumerative meanings given in the source. The open-loop 1-sum conjecture.

$$S_{2n}^{(1)}=A_{\rm v}(2n+1), \qquad S_{2n-1}^{(1)}=N_8(2n).$$

This is an O(1) loop-model analogue of the XXZ-chain sum formulas and connects the sums to vertically symmetric alternating sign matrices and cyclically symmetric transpose-complement plane partitions. The source offers finite-size evidence but no proof.