Let $R$ be a Dedekind domain with fraction field $K$, and let $G$ be a smooth affine group scheme over $S = \text{Spec }R$ whose geometric fibers are connected and simple linear algebraic groups (i.e., $G$ is a "connected simple linear algebraic $S$-group scheme").

Does $G$ admit a universal cover over $S$? Here, by universal cover I mean a finite flat surjective homomorphism of $S$-group schemes $p : \tilde{G}\rightarrow G$ such that $p$ induces universal covers on every geometric fiber in the usual sense.

If not, can we always do this Zariski or etale locally on $S$?

I think this is true if $S = \text{Spec }k$ with $k$ a field, though I don't know of a reference. References are also appreciated!

EDIT: Let me elaborate on what I mean by universal cover. I believe split reductive groups all exist as smooth group schemes over $\mathbb{Z}$ (with reductive geometric fibers), and they are uniquely determined up to isomorphism by their $\mathbb{C}$-geometric fiber. Also I believe the isogeny theorem (Theorem 6.1.16 of Brian Conrad's notes) should imply that a universal cover $\tilde{\mathbf{G}}\rightarrow \mathbf{G}$ of simple algebraic groups over $\mathbb{C}$ induces an isogeny of their unique split $\mathbb{Z}$-models $\tilde{G}\rightarrow G$. I want this latter map and any of its base changes to be considered a universal cover. If $k$ is an algebraically closed field of characteristic $p$, then I think I want a universal cover of a simple linear algebraic $k$-group $G$ to be an isogeny from a simple linear algebraic $k$-group $\tilde{G}$ such that any central isogeny onto $\tilde{G}$ is trivial. Please correct me if I'm wrong!

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