On the length of reduced Gröbner bases

Question

Say $I\subset J$ are two ideals of the ring $k[x_1,...,x_n]$ where $k$ is an algebraically closed field and $G,H$ are the reduced Gröbner bases of $I$ and $J$ respectively (for some monomial ordering). Is it true that $|G|\leq |H|$? I believe that it holds for at least prime ideals but I don't have a proof.

Answer 1

For the case of non-prime ideals, I see now that the answer is an easy "no".

Here's a counterexample . . .

In the ring $k[x,y]$, where $k$ is any field, consider the ideals $I,J$, where $$I = (x^2,xy,y^2),\;\;J = (x,y)$$ Then for any monomial ordering, the ideals $I,J$ are Groebner reduced.