Erdős's bounded finite-sums conjecture for positive-density sets

Let $A oりsubset\mathbb{N}$ have positive upper Banach density. For $k\in\mathbb{N}$, write

$$\operatorname{FS}_{\leq k}(B)=\left\{\sum_{n\in F}n:F\subset B,\ 0<|F|\leq k\right\}.$$

Erdős's conjecture. For every $k\in\mathbb{N}$, there exist an integer $t\geq0$ and an infinite set $B\subset\mathbb{N}$ such that

$$\operatorname{FS}_{\leq k}(B)\subset A-t.$$

The case $k=2$ is proved, while the conjecture remains open for $k\geq3$. The claim is a density version of Hindman's theorem and is designed to avoid known obstructions to unrestricted sumsets.