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., $$B=-B$$)?