Consider the monomial ideal $I=(x^d, y^az^{d-a})$ in $\mathbb C [x,y,z]$ where $1\le a\le d-1$ are integers. Let $\mathfrak m=(x,y,z)$. Let $J=\overline I$ be the integral closure of $I$.
If $J\mathfrak m$ is not integrally closed, then is it true that $(\overline {J+\mathfrak m^t })\mathfrak m$ is not integrally closed for some integer $t\ge 1$ ?
I think either $t=d+1$ or $t=d+2$ or $t=d+3$ should work, but I'm not sure.
The motivation for my question is: from the knowledge of an integrally closed monomial ideal $J$ such that $J\mathfrak m$ is not integrally closed, I'm trying come up with an integrally closed monomial $\mathfrak m$-primary ideal $L$ such that $L\mathfrak m$ is not integrally closed.
Please help.