Let $R=\mathbb C[x,y]$. Let $\mathfrak m=(x,y)$ . Let $J \subseteq \mathfrak m$ be a homogenous ideal with $\sqrt J=\mathfrak m$ i.e. $\mathfrak m^n \subseteq J$ for some integer $n\ge 1$.
Let $a\ge 1$ be the smallest integer such that $J \cap \mathfrak m^{a+1} \subseteq J\mathfrak m$ and let $b\ge 1$ be the smallest integer such that $(J \cap \mathfrak m^b)\mathfrak m^{bs}=\mathfrak m^{bs+b}$ for some integer $s\ge 0$.
Then is it true that $b \le a$ ? Is it at least true when $J$ is a monomial ideal ?
[Note that of-course $a$ and $b$ depends on $J$]