Let $S$ be a scheme, $G$ a smooth $S$-affine $S$-group scheme, and $X$ an $S$-scheme with a ($S$-linear) $G$-action $\alpha : G\times X\rightarrow X$.

Let $h_X$, $h_G$ be the representable sheaves on the small étale site $S_\text{ét}$ associated to $X,G$ respectively. Suppose $h_X$ is a torsor under $h_G$ (i.e., $h_G(T)$ acts simply transitively on $h_X(T)$ for every étale $T\rightarrow S$). Must $X$ be a $G$-torsor? (i.e., $\alpha\times\mathrm{id} : G\times X\rightarrow X\times X$ is an isomorphism).

If necessary, I'm happy to assume $G$ reductive and that $X$ is $S$-smooth and $S$-affine.

If this is true, a pleasing consequence is that it suffices to check that $h_G(\operatorname{Spec} R)$ is simply transitive on $h_X(\operatorname{Spec} R)$ for strict local rings $R$ of $S$.