Is the following true?

CONJECTURE: $\,$ Let $\ B\ C\subseteq\mathbb R^n\ $ be convex bodies such that $\ C\ $ is centrally symmetric, $\ B\subseteq C,\ $ and $\ t\!\cdot\! B\ $ cannot be isometrically embedded in $\ C,\ $ for no $\ t>1.\ $ Then $\ c(C)\in B$.

Thank you Douglas Z. for pointing out the mess in my earlier formulation.