Density bound conjecture for polynomial cube progressions

Let $P_1,\dots,P_s\in\mathbb{Z}[x]$ be linearly independent polynomials that vanish at zero. For $\omega\in\{0,1\}^s$, write $\omega=(\omega_1,\dots,\omega_s)$ and $\omega\cdot P(y)=\sum_{i=1}^s\omega_iP_i(y)$. Polynomial progression density conjecture. If a subset $A\subseteq\mathbb{Z}/N\mathbb{Z}$ lacks the progression

$$(x+\omega\cdot P(y))_{\substack{\omega\in\{0,1\}^s\\|\omega|\le t}}$$

with $y$ nonzero, then

$$|A|\le O\left(\frac{N}{\log_{O(1)}(N)}\right).$$

This is stated as a corollary that would follow from the preceding conjectural estimate, giving a quantitative Szemerédi-type density bound for these polynomial patterns; it is not proved in the paper.