The ordered linear sumset conjecture

Let $A\subset\mathbb{N}$ have positive upper Banach density, and let $\ell,m\in\mathbb{N}$. For an increasing sequence $B=\{b_1<b_2<\cdots\}$, consider the ordered linear configuration

$$\{\ell b_i+m b_j:i<j\}.$$

Ordered linear sumset conjecture. There exist an infinite increasing set $B=\{b_1<b_2<\cdots\}\subset\mathbb{N}$ and an integer $t\geq0$ such that

$$\{\ell b_i+m b_j:i<j\}\subset A-t.$$

This is the linear case of the paper's ordered polynomial sumset questions. The source presents it as a conjecture motivated by partition regularity, while examples show that the ordering restriction matters.