Let $G$ be a torsion free group with identity $e$. For a subset $X$ of $G$, denote by $X^\#$ the set $X\setminus\{e\}$. Let $A$ be a finite subset of $G$ containing $e$. Is there a finite subset $B$ containing $e$ such that $$A\subset B^\#A\quad\text{and}\quad B\subset BA^\#$$ It is obvious there is no such $B$ when $A=\{e,x\}$; this follows from the right inclusion and that $G$ is torsion free, so the $A$ in my question has at least 3 elements.

The above question comes from the zero divisor conjecture; if there exist non zero $\alpha,\beta\in\mathbb C[G]$ such that $\alpha\beta=0$, without loss of generality we can assume $A:=\text{support}(\alpha)$ and $B:=\text{support}(\beta)$ contain $e$, and then $A$ and $B$ should satisfy in above inclusions.