The Type BII first-coefficient conjecture

At $q=1$, write the Type BII sum as

$$S_N=S^{\mathrm{BII}}_N=\sum_{j=0}^{N}S_{N,j}(Q+Q^{-1})^j,$$

with $S_{N,j}\in\mathbb{N}$. Type BII first-coefficient conjecture.

$$S_{N,1}=\frac{1}{8}\left(2N^2+4N+1-(-1)^N\right).$$

The paper reports verification through $N=20$, but no general proof or resolution is supplied.