Is the following true?

**CONJECTURE:** $\,$ Let $\ B\ C\subseteq\mathbb R^n\ $ be convex bodies such that $\ C\ $ is centrally symmetric, and

$$ B\subseteq C\ \nsubseteq\ t\!\cdot\! B $$

for arbitrary $t>1$. Is $\ c(C)\in B,\ $ where $\ c(C)\ $ is the center of $C$?