Inverse theorem conjecture for polynomial progression averages

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, and define

$$\Lambda_{s,t}(f_\omega)=\mathbb{E}_{x,y}\prod_{\substack{\omega\in\{0,1\}^s\\|\omega|\le t}}f_\omega(x+\omega\cdot P(y)).$$

Inverse theorem conjecture. If

$$|\Lambda_{s,t}(f_\omega)|\ge\delta,$$

then either

$$\delta\le\frac{1}{\log_{O(1)}(N)}$$

or

$$\|f_\omega\|\ge\frac{1}{\exp_{O(1)}(\delta^{-1})}.$$

This is intended as a quantitative inverse statement for the polynomial progression average, extending the methods and results discussed in the paper; it remains unproved in the supplied text.