Generalized sumset pattern conjecture for multiple uniform sets

Let $A_1,\ldots,A_k\subset\mathbb N$ be $U^k(\Phi)$-uniform sets for the same Følner sequence $\Phi$. Let $\ell$ be a positive integer and choose integers

$$1\leq \ell_1<\ell_2<\cdots<\ell_k\leq \ell.$$

Generalized sumset pattern conjecture. There exists an infinite set $B\subset\mathbb N$ such that, for every $i=1,\ldots,k$,

$$B^{\oplus \ell_i}\subset A_i.$$

This conjecture generalizes the corresponding result for a single uniform set and asks for a common infinite additive pattern across multiple $U^k(\Phi)$-uniform sets. The source presents the statement as an open question; the uniformity theorem mentioned there is a special case.