A local ring such that the map between Grothendieck groups with rational coefficient induced by completion is not injective

In this paper, we construct a local ring $A$ such that the kernel of the map $G_0(A)\text{subq}\to G_0(\widehat{A})\text{subq}$ is not zero, where $\widehat{A}$ is the comletion of $A$ with respect to the maximal ideal, and $G_0()\text{subq}$ is the Grothendieck group of finitely generated modules with rational coefficient. In our example, $A$ is a two-dimensional local ring which is essentially of finite type over $\mathbb{C}$, but it is not normal.