Let $R$ be an integral $\bar{k}$-algebra of finite type. Let $V(I) \subseteq \mathbb{A}_R^n$ be a reduced (closed) subgroup scheme such that $V(I)\backslash \{0\} \neq \emptyset$ and the $\mathbb{G}_m^R$-action on $\mathbb{A}_R^n \backslash \{0\}$ restricts to a free action on $V(I)\backslash \{0\}$. Is it true that $I$ can generically be generated by linear polynomials? (If $R=k$ is any field, and $V(I)$ as above, can $I$ be generated by linear polynomials?)

If $R=\bar{k}$ this is clear, because we can just check it on $\bar{k}$-points (should not even need the $\mathbb{G}_m^R$-action). In general, intuitively this seems good to me (I would just think of 'generalised' subvector spaces), but I know too few about group schemes to verify it. I guess if there was a counterexample, then it would be in the non-perfect world.

Edit: As Jason pointed out there are indeed counterexamples. What if we additionally assume that $V(I)$ has reduced (closed) fibers?