Bosch–Lütkebohmert–Raynaud's conjecture on Néron models of smooth algebraic groups

Let $S$ be a global excellent Dedekind scheme with function field $K$, and let $G$ be a smooth algebraic group over $K$. A Néron lft-model of $G$ is a smooth separated $S$-group algebraic space, locally of finite type, with generic fibre $G$ and the Néron mapping property; a Néron model is a Néron lft-model that is of finite type over $S$. A subgroup is unirational if its underlying scheme admits a scheme-theoretically dominant morphism from an open subscheme of some affine space over $K$. Bosch–Lütkebohmert–Raynaud's conjecture. (I) If $G$ contains no subgroup isomorphic to $\operatorname{\mathbf{G}_a}$, then $G$ admits a Néron lft-model over $S$. (II) Moreover, if $G$ contains no non-trivial unirational closed subgroup, then $G$ admits a Néron model over $S$. The conjecture proposes global criteria for the existence, respectively finite-type existence, of Néron models, extending the completely understood local case. The source gives no resolution of either assertion.