Grothendieck–Serre conjecture on principal bundles over regular local rings

Let $R$ be a regular local ring, let $K$ be its field of fractions, and set $U:=\operatorname{Spec} R$. Let $\mathbf G$ be a reductive group scheme over $U$, and let $\mathcal G$ be a principal $\mathbf G$-bundle. If $\mathcal G$ is trivial over $\operatorname{Spec} K$, then it is trivial. Grothendieck–Serre conjecture. Equivalently, the map of non-abelian cohomology pointed sets

$$H^1_{\text{\'et}}(R,\mathbf G)\to H^1_{\text{\'et}}(K,\mathbf G)$$

induced by the inclusion of $R$ into $K$ has a trivial kernel. The paper proves this conjecture for regular semi-local domains containing a finite field; together with the result for domains containing an infinite field, this establishes the conjecture for semi-local regular domains containing a field.