Existence of a symmetric subset $B\subseteq A$ such that $2A-A\subseteq 8A$

Let $ A$ be a nonempty open connected subset of a (real) topological vector space $ X$ such that $ $ 2A-A \subseteq 8A$ $ (for instance one could take $ A=(-1,2)$ ).

Question. Is it true that there exists a nonempty open connected set $ B\subseteq A$ such that $ B$ , in addition, is symmetric (i.e., $ B=-B$ )?