Is it true that every closed ideal $J$ of $ A \otimes_hB$ is of the form $\epsilon^{-1}(I)$ for some closed ideal $I$ of $A \otimes_\text{min} B$.

Question

Let $A$ and $B$ be $C^{\ast}$-algebras. It is known that the map $\epsilon:A \otimes_h B \to A \otimes_\text{min} B$ is continuous and injective.

Is it true that every closed ideal $J$ of $ A \otimes_hB$ is of the form $\epsilon^{-1}(I)$ for some closed ideal $I$ of $A \otimes_\text{min} B$.

Setting $I= \overline{\epsilon(J)}$ works?

P.S: Copy of the same question on MO can be found here.