Let $k$ be a field and $A$ a noetherian local $k$-algebra. Let $I$ and $J$ be two ideals of $A$ with $I^2 \subset J \subset I$. Let $A[tI] = \bigoplus\limits_{i\geq 0} t^i I^i$ and $A[tJ] = \bigoplus\limits_{i\geq 0} t^i J^i$ be the Rees algebras for $I$ and $J$, respectively.

**Question:** Is it true that $\mathrm{Proj}(A[tI]) \cong \mathrm{Proj}(A[tJ])$ as $k$-schemes?
If not, then in what condition one can get such an isomorphism of $k$-schemes?