Monochromatic finite-sums and product-of-sums conjecture

Let $\mathbb{N}=\bigcup_{i=1}^m A_i$ be a finite partition, let $\mathcal{P}_f(\mathbb{N})$ denote the nonempty finite subsets of $\mathbb{N}$, and let $\alpha<\beta$ mean $\max\alpha<\min\beta$. Finite-sums and product-of-sums conjecture. There exist $j\in[m]$, $b>1$, and a sequence $\langle x_t\rangle_{t\in\mathbb{N}}$ such that

$$\left\{\sum_{t\in\alpha}x_t,\ \prod_{s=1}^b\sum_{t\in\alpha_s}x_t:\ \alpha\in\mathcal{P}_f(\mathbb{N}),\ \alpha_1<\alpha_2<\dots<\alpha_b\in\mathcal{P}_f(\mathbb{N})\right\}\subseteq A_j.$$

Thus one cell of every finite partition should contain all finite sums from one sequence together with the products of $b$ sums over successive finite blocks, for some $b>1$. The question is stated as open, and an affirmative answer would follow from the paper's return-time conjecture.