Higher-order shifted finite-sums conjecture

Let $A\subset\mathbb{N}$ have positive upper Banach density, and let $k\in\mathbb{N}$. For a finite subset $F$ of $B$, write its sum as $\sum_{n\in F}n$. Higher-order shifted finite-sums conjecture. There exist an infinite set $B\subset A$ and a shift $t\in\mathbb{N}$ such that

$$A-t\supset \left\{\sum_{n\in F}n:F\subset B,\ 0<|F|<k\right\}.$$

This is presented as a natural conjectural higher-order analogue of the paper's main theorem, replacing two-term restricted sums by sums over finite subsets of bounded cardinality. The supplied text gives no resolution or evidence that the claim is proved or disproved, so its status remains open.