Let $R$ be a local integral domain with residue field $k$ such that $R$ is of characteristic zero and $k$ is of characteristic $p>0$. Let $G$ be a smooth finite type affine group scheme with geometrically connected fibres over $R$. Suppose that $T$ is a $G_k$-torsor over $k$.

Does $T$ lift to $R$?

That is, does there exist a $G$-torsor $\mathcal T$ over $R$ such that $\mathcal T \otimes_R k$ is isomorphic to $T$ over $k$?

If so, can we then show the stronger assertion that the map on cohomology sets $H^1_{et}(R,G_R)\to H^1_{et}(k,G_k)$ is surjective for all complete local regular rings $R$ with residue field $k$?

Motivation: I'm trying to understand what it means for a variety over $k$ to lift to characteristic zero. I know curves, ppav's, K3 surfaces and hypersurfaces lift to characteristic zero. I also know examples of Fano varieties, Calabi-Yau threefolds and some surfaces of general type which don't lift to characteristic zero. Out of curiosity I wondered what one can say about liftability of torsors (under affine group schemes). I hope the comments and answers will shed some light on this matter for me.