The Type BIII component signed-permutation formula

Let $\mathcal{C}_{N+1}$ be the set of signed permutation matrices of size $2N\times2N$ invariant under diagonal and anti-diagonal reflections and avoiding $(-2,-1)$. Let $B(c)$ be the binary-string map defined by the down-left paths from fundamental links to diagonal points, and let $D(b)$ be the corresponding Type BIII diagram. Type BIII component formula. At $q=Q=1$,

$$\Psi_{D(b)}=\#\{c\in\mathcal{C}_{N+1}\mid B(c)=b\}.$$

This is a componentwise combinatorial interpretation of the ground state; the source presents it as a conjecture and supplies no proof or resolution.