Proof of the Grothendieck–Serre conjecture on principal bundles over regular local rings containing a field

Let $R$ be a regular local ring containing a field. Let $\mathbf{G}$ be a reductive group scheme over $R$. We prove that a principal $\mathbf{G}$-bundle over $R$ is trivial if it is trivial over the field of fractions of $R$. In other words, if $K$ is the field of fractions of $R$, then the map
$$ H^1_{\mathrm{et}}(R,\mathbf{G})\to H^1_{\mathrm{et}}(K,\mathbf{G}) $$
of the non-Abelian cohomology pointed sets induced by the inclusion of $R$ in $K$ has trivial kernel. This result was proved in [1] for regular local rings $R$ containing an infinite field.