On the Grothendieck–Serre conjecture concerning principal $G$-bundles over semi-local Dedekind domains

Let $R$ be a semi-local Dedekind domain and let $K$ be the field of fractions of $R$. Let $G$ be a reductive semisimple simply connected $R$-group scheme such that every semisimple normal $R$-subgroup scheme of $G$ contains a split $R$-torus $\mathbb G_{m,R}$. We prove that the kernel of the map
$$ H^1_{\unicode{x00E9}\unicode{x74}}(R,G)\to H^1_{\unicode{x00E9}\unicode{x74}}(K,G) $$
induced by the inclusion of $R$ into $K$, is trivial. This result partially extends the Nisnevich theorem [10, Thm.4.2].