Self-dual sets up to a translation: a negative answer to Milman's question

Let $K,T\subset {\mathbb{R}}^n$ be convex bodies with the origin in their interiors. If $K=T^{\circ }$, then the identity $K+T=K^{\circ }+T^{\circ }$ holds trivially. V.~Milman asked whether these trivial solutions are the only ones, that is, whether $K+T=K^{\circ }+T^{\circ }$ forces $K=T^{\circ }$. We show that the answer is positive for polytopes but negative for general convex bodies.