The Type BII coefficient-polynomial conjecture

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

$$S_N=\sum_{j=0}^{N}S_{N,j}(Q+Q^{-1})^j.$$

Type BII coefficient-polynomial conjecture. For the relevant coefficient index $i$,

$$S_{N,N-i}=\left(\prod_{j=1}^{i}(2j)^{-1}\right)P_i(N),$$

where

$$P_i(N)=N^{2i}-iN^{2i-1}+\sum_{k=0}^{2i-2}p_{i,k}N^k,$$

with $p_{i,k}\in\mathbb{Z}$. The source reports verification through $j=7$ and $N=20$ and gives initial polynomials, but no proof or resolution is supplied.