The group-theoretic two-fold sumset conjecture

Let $(G,+)$ be a countable abelian group, and let $A\subset G$ have positive upper Banach density. For an infinite set $B\subset G$, define

$$B\oplus B=\{b+b':b,b'\in B,\ b\ne b'\}.$$

Group-theoretic sumset conjecture. There exist an infinite set $B\subset G$ and $t\in G$ such that

$$B\oplus B+t\subset A.$$

The source asks broader questions about sumsets in countable abelian groups and records this as the corresponding conjectural two-fold configuration. Positive results are suggested for $\mathbb Z^d$, but no general resolution is supplied.