On asymptotic depth of integral closure filtration and an application

Let $(A,mathfrak{m})$ be an analytically unramified formally equidimensional Noetherian local ring with $ depth A geq 2$. Let $I$ be an $mathfrak{m}$-primary ideal and set $I^*$ to be the integral closure of $I$. Set $G^*(I) = bigoplus_{ngeq 0} (I^n)^*/(I^{n+1})^*$ be the associated graded ring of the integral closure filtration of $I$. We prove that $ depth G^*(I^n) geq 2$ for all $n gg 0$. As an application we prove that if $A$ is also an excellent normal domain containing an algebraically closed field isomorphic to $A/m$ then there exists $s_0$ such that for all $s geq s_0$ and $J$ is an integrally closed ideal emph{strictly} containing $(mathfrak{m}^s)^*$ then we have a strict inequality $mu(J) < mu((mathfrak{m}^s)^*)$ (here $mu(J)$ is the number of minimal generators of $J$).