Quantitative generalized von Neumann conjecture for polynomial progressions

Let $P_1,\dots,P_s\in\mathbb{Z}[x]$ be linearly independent polynomials that vanish at zero. Let $f_\omega:\mathbb{Z}/N\mathbb{Z}\to\mathbb{C}$ be one-bounded functions for each $\omega\in\{0,1\}^s$, and let $0\le t\le s$ be a natural number. Write

$$P(y)=(P_1(y),P_2(y),\dots,P_s(y)),\qquad \omega=(\omega_1,\dots,\omega_s),$$

and define $\omega\cdot P(y)=\sum_{i=1}^s\omega_iP_i(y)$. Quantitative polynomial progression conjecture. One has

$$\mathbb{E}_{x,y}\prod_{\substack{\omega\in\{0,1\}^s\\|\omega|=t}}f_\omega(x+\omega\cdot P(y)) =\mathbb{E}_{x,h_1,\dots,h_s}\prod_{\substack{\omega\in\{0,1\}^s\\|\omega|\le t}}f_\omega(x+\omega\cdot h)+O\left(\frac{1}{\log_{(O(1))}(N)}\right).$$

This would extend the paper's quantitative counting result to polynomial patterns of true complexity one and to other progressions considered in the cited work; the claim is presented as a direction for further work and is not established here.