The uniform sets bounded-sums conjecture

Let $A\subset\mathbb{N}$ and $k\geq2$. Let $\Phi=(\Phi_N)_{N\in\mathbb{N}}$ be a Følner sequence. A set is $\mathsf{U}^k(\Phi)$-uniform when its indicator has correlations along $\Phi$, positive density along $\Phi$, and vanishing $\mathsf{U}^k(\Phi)$ seminorm after subtracting its density. For an infinite $B\subset\mathbb{N}$ and $\ell\in\mathbb{N}$, write

$$B^{\oplus\ell}=\left\{\sum_{n\in F}n:F\subset B,\ |F|=\ell\right\}.$$

Uniform sets bounded-sums conjecture. If $A$ is $\mathsf{U}^k(\Phi)$-uniform for some Følner sequence $\Phi$, then for every $\ell_1,\ldots,\ell_k\in\mathbb{N}$ there is an infinite set $B\subset\mathbb{N}$ such that

$$B^{\oplus\ell_1},B^{\oplus\ell_2},\ldots,B^{\oplus\ell_k}\subset A.$$

The conjecture proposes that the relevant uniformity hypotheses remove the shift required in the general positive-density problem. It is stated without a resolution in the supplied text.