Ideals of $A\otimes\mathcal{C}_0(0,1]$ for $A$ a simple unital $C^*$ algebra

Question

Let $A$ be a simple, unital, separable $C^*$-algebra and $J$ a closed two-sided ideal of $A \otimes\mathcal{C}_0(0,1]$, is it true that $J=A\otimes\mathcal{C}_0(\mathcal{O})$ for some open subset of $(0,1]$?