The Type BIII signed-permutation coefficient conjecture

Let $\mathcal{C}_n$ be the set of signed permutation matrices of size $2(n-1)\times2(n-1)$ invariant under diagonal and anti-diagonal reflections and avoiding the pattern $(-2,-1)$. Define the weight of $c$ as in the source from the difference between the specified positive and negative regions, and write the Type BIII sum at $q=1$ as

$$S_N=\sum_{i=0}^{N}\widetilde{S}_{N,N-2i}Q^{N-2i}.$$

Type BIII signed-permutation conjecture.

$$\widetilde{S}_{N,i}=\#\{c\in\mathcal{C}\mid \operatorname{wt}(c)=i\}.$$

This predicts a weight-preserving signed-permutation interpretation of the coefficients, but the source supplies no proof or resolution.