A solution to Itoh's conjecture for integral closure filtration

Let $(A,\mathfrak{m})$ be an analytically unramified Cohen-Macaulay local ring of dimension $d\ge 3$ and let $\mathfrak{a}$ be an $\mathfrak{m}$-primary ideal in $A$. If $I$ is an ideal in $A$ then let $I^{\ast }$ be the integral closure of $I$ in $A$. Let $G_{\mathfrak{a}}(A{)}^{\ast }={⨁}_{n\ge 0}({\mathfrak{a}}^n{)}^{\ast }\mathrm{/}({\mathfrak{a}}^{n+1}{)}^{\ast }$ be the associated graded ring of the integral closure filtration of $\mathfrak{a}$. Itoh conjectured in 1992 that if third Hilbert coefficient of $G_{\mathfrak{a}}(A{)}^{\ast }$ , i.e., $e_3^{{\mathfrak{a}}^{\ast }}(A)=0$ and $A$ is Gorenstein then $G_{\mathfrak{a}}(A{)}^{\ast }$ is Cohen-Macaulay. In this paper we prove Itoh’s conjecture (more generally for analytically unramified Cohen-Macaulay local rings).