The Type A symmetric-binary-matrix coefficient conjecture

Let $\mathcal{A}_N$ be the set of symmetric binary $N\times N$ matrices with no row sum greater than one. For $a=(a_{i,j})$, define

$$\operatorname{wt}(a)=\#\{a_{i,j}=1\mid i\leq j\}.$$

Let $S_{N,i}$ be the coefficient of $Q^i$ in the Type A sum at $q=1$. Type A matrix conjecture.

$$S_{N,i}=\#\{a\in\mathcal{A}_N\mid \operatorname{wt}(a)=i\}.$$

This refines the total-sum enumeration by predicting a weight-preserving interpretation of every coefficient; the source does not provide a proof or resolution.