The quantum-invariant XXZ-chain 1-sum conjecture

Consider the quantum-invariant open XXZ chain at $\Delta=-\frac{1}{2}$, with groundstate wavefunction normalised as in the source. Let $S_L^{(1)}$ denote the sum of its components. Define $A_{\rm v}(2n+1)$ by

$$A_{\rm v}(2n+1)=(-3)^{n^2}\prod_{i=1}^{2n+1}\prod_{j=1}^{n}\frac{3(2j-i)+1}{2j-i+2n+1}$$

and $N_8(2n)$ by

$$N_8(2n)=\prod_{i=1}^{n-1}(3i+1)\frac{(2i)!(6i)!}{(4i)!(4i+1)!}.$$

The quantum-invariant-chain 1-sum conjecture.

$$S_{2n}^{(1)}=3^{n/2}A_{\rm v}(2n+1), \qquad S_{2n-1}^{(1)}=3^{(n-1)/2}N_8(2n).$$

Here $A_{\rm v}(2n+1)$ counts vertically symmetric alternating sign matrices and $N_8(2n)$ counts cyclically symmetric transpose-complement plane partitions. The formulas are conjectured from finite-size data and are unresolved in the source.