Relative compactification conjecture for semiabelian Néron models

Let $R$ be a complete discrete valuation ring with fraction field $k(\eta)$, let $(G_{\eta},{\mathcal L}_{\eta})$ be a polarized abelian variety over $k(\eta)$, let ${\mathcal G}$ be the Néron model of $G_{\eta}$, let $G:={\mathcal G}^0$ be its identity component, and set $N:=|{\mathcal G}_0/G_0|$. Choose an ample cubical invertible sheaf ${\mathcal N}$ on ${\mathcal G}$ with ${\mathcal N}_{\eta}\simeq{\mathcal L}_{\eta}^{\otimes 2N}$, and define

$$P_n:=\operatorname{Proj} A({\mathcal G},{\mathcal N}^{\otimes n}).$$

Relative compactification conjecture. For some $n_0\in{\mathbf N}$, $P_{n_0}$ is a relative compactification of ${\mathcal G}$ and

$${\mathcal G}\simeq P_{n_0}\setminus\operatorname{Sing}(P_{n_0}/S),\qquad \operatorname{codim}_{P_{n_0}}\operatorname{Sing}(P_{n_0}/S)\geq 2.$$

This asserts the existence of a projective relative compactification obtained from the graded algebra associated with a suitable tensor power of the cubical line bundle, while identifying the semiabelian Néron model with the complement of the relative singular locus. The supplied span gives no resolution status, so the claim is recorded as open.