Stabilization conjecture for bow-variety generating-series coefficients

For $e\in \mathbb{Z}^n$ and $f\in \mathbb{Z}^m$, let

$$Z(q,t)=\sum_{s=0}^{\infty}P_s(t)q^s.$$

Here $P_s(t)$ is a polynomial, and $N_s$ is a sequence of integers with $N_s\to\infty$. Stabilization conjecture. The polynomial $P_s(t)$ agrees with the series

$$\prod_{r=1}^{\infty}\frac{1}{(1-t^{2r})^m}$$

up to degree $N_s$. This predicts stabilization of the low-degree coefficients of the generating series as $s$ increases. The statement is presented as a conjectural generalization of the stabilization observed in examples; no resolution is given here.