Let $(R,\mathfrak m)$ be a local Gorenstein ring (may also assume excellent) of dimension at least $2$, and let $(\hat R,\hat {\mathfrak m})$ be the $\mathfrak m$-adic completion. Let $U:=\text{Spec}(R)\setminus \{\mathfrak m\}$ be the punctured spectrum of $R$ and and $\hat U:=\text{Spec}(\hat R)\setminus \{\hat {\mathfrak m}\}$ be the punctured spectrum of $\hat R$.
So we have a canonical map $K_0(U)\to K_0(\hat U)$ (where $K_0(-)$ denotes the Grothendieck group of vector-bundles) . My question is: Is this map necessarily injective ? If this is not true in general, what conditions on $R$ would make it true ?