# Ideals with equal squares in a Noetherian UFD of dimension 2

Let $$I$$ and $$J$$ be two ideals in $$\mathbb C[X,Y]$$ such that $$\mu(I)=\mu(J) \le 3$$ and $$I^2=J^2$$ . Then is it necessarily true that $$I=J$$ ? If not, then is it at least true if we assume $$I,J$$ are homogeneous ideals ?

I believe it should not be true but I am having a hard time finding a counterexample.