Gorenstein Normal tangent cones of integrally closed ideals in two-dimensional normal singularities

Let $(A,\mathfrak{m})$ be a two-dimensional excellent normal Gorenstein local domain containing an algebraically closed filed. Let $I=H^0(X,{\mathcal{O}}_X(-Z))\subset A$ be an $\mathfrak{m}$-primary integrally closed ideal represented by an anti-nef cycle $Z$ on some resolution $X\to \mathrm{S}\mathrm{p}\mathrm{e}\mathrm{c}A$. In this paper, we prove that $\bar{G}(I)$ is Gorenstein if and only if it is Cohen-Macaulay and $(r-1)Z^2+K_{X}Z=0$, where $r=\overline{\text{athrm}r}(I)$ denotes the normal reduction number of $I$ and $K_X$ denotes the canonical divisor on $X$.