The finite-size component conjecture for the supersymmetric open XXZ chain

For each $n\geqslant 1$, let $|\psi_L\rangle$ be the zero-energy state of the open XXZ Hamiltonian, and let $A_{\text{V}}(2n+1)$ and $N_8(2n)$ denote respectively the numbers of vertically symmetric alternating sign matrices of size $(2n+1)\times(2n+1)$ and cyclically symmetric self-complementary plane partitions in a $2n\times 2n\times 2n$ cube. They are given by

$$A_{\text{V}}(2n+1)=\frac{1}{2^n}\prod_{k=1}^n\frac{(6k-2)!(2k-1)!}{(4k-1)!(4k-2)!},\qquad N_8(2n)=\prod_{k=0}^{n-1}\frac{(3k+1)(6k)!(2k)!}{(4k)!(4k+1)!}.$$

Finite-size component conjecture. For each $n\geqslant 1$,

$$\frac{(\psi_{2n-1})_{01\cdots 010}}{\lVert\psi_{2n-1}\rVert}=\sqrt{\frac{N_8(2n)}{A_{\text{V}}(2n+1)}},\qquad \frac{(\psi_{2n})_{01\cdots 01}}{\lVert\psi_{2n}\rVert}=\sqrt{\frac{A_{\text{V}}(2n+1)}{N_8(2n+2)}}.$$

These formulas were inferred from exact diagonalisation through $L=16$ sites and connect special ground-state components with enumerative-combinatorial sequences. Their validity for all $n$ remains unproved in the source.