The coefficient-polynomial conjecture for Type A and Type BIII sums

At $q=1$, write the Type A and Type BIII sums as

$$S^{\mathrm{A}}_{N}=\sum_{i=0}^{N}S_{N,i}Q^{i},\qquad S^{\mathrm{BIII}}_{N}=\sum_{i=0}^{N}S'_{N,i}(Q+Q^{-1})^{i},$$

with nonnegative integer coefficients $S_{N,i}$ and $S'_{N,i}$. Coefficient-polynomial conjecture. The coefficients satisfy

$$S_{N,i}=S_{N,N-i}=S'_{N,i}=S'_{N,N-i},$$

and

$$S_{N,i}=\left(\prod_{k=0}^{i-1}\frac{N-k}{2k+2}\right)P_i(N),\qquad P_i(N)=N^i+\sum_{j=0}^{i-1}p_{i,j}N^j,$$

where $p_{i,j}\in\mathbb{Z}$. The source presents this as an empirical conjecture and lists the first few polynomials; no proof or resolution is given.