Let $I=(f,g,h)$ be an ideal in the polynomial ring $k[x,y,z]$ with $LT(f)>LT(g)>LT(h)$ in the lexorder, and $I$ is "reduced" in the sense that $LT(g)\nmid LT(f),LT(h)\nmid LT(g),LT(h)\nmid LT(f)$. Is it necessary that the reduced Groebner basis $G$ of $I$ has at least $3$ elements?
Edit: We may restrict to the nontrivial case.