Let $(R, \mathfrak m)$ be an excellent, normal, local domain of dimension $2$ containing an algebraically closed field $k=R/\mathfrak m$. Let $ \pi: Y \to X=\operatorname {Spec}(R)$ be a resolution of singularities (which exists by a Theorem of Lipman) and also assume $H^1(Y, \mathcal O_Y)=0$ i.e. $X$ has geometric genus zero (this doesn't depend on the choice of resolution). One Algebraic characterization of the vanishing of this sheaf Cohomology in this context is that $IJ$ is an integrally closed ideal for every integrally closed ideals $I$ and $J$ of $R$ (See Theorem 1 of https://doi.org/10.1007/BF01233425)
Let $G_0(R)$ be the Grothendieck group of the abelian category of finitely generated $R$-modules.
Then, is it true that $G_0(R)\otimes_{\mathbb Z} \mathbb Q$ is a finite dimensional $\mathbb Q$-vector space ?