The equivalence between torus and reductive Grothendieck–Serre injectivity

Let $R$ be an integral domain with fraction field $K$. For a group scheme ${\mathbf G}$ over $R$, write $H^1(R,{\mathbf G})$ and $H^1(K,{\mathbf G})$ for the corresponding nonabelian cohomology sets.

Torus–reductive injectivity equivalence. The following conditions are equivalent:

1. Every vector bundle over $\operatorname{Spec} R$ is trivial, and for every torus ${\mathbf T}$ the map

$$H^1(R,{\mathbf T})\to H^1(K,{\mathbf T})$$

has trivial kernel. 2. For every reductive $R$-group scheme ${\mathbf G}$, the map

$$H^1(R,{\mathbf G})\to H^1(K,{\mathbf G})$$

has trivial kernel.

This characterizes the Grothendieck–Serre injectivity property for all reductive group schemes in terms of the corresponding property for tori together with triviality of vector bundles. The supplied text does not state whether this equivalence is proved or remains open.