Toric geometry and the standard conjecture for a compactification of the Néron model of Abelian variety over $1$-dimensional function field

It is proved that if $\mathcal M\to C$ is the Néron minimal model of a principally polarized $(d-1)$-dimensional Abelian variety $\mathcal M_\eta$ over the field $\kappa(\eta)$ of rational functions of a smooth projective curve $C$,
$$ \operatorname{End}_{\overline{\kappa(\eta)}} (\mathcal M_\eta\otimes_{\kappa(\eta)}\overline{\kappa(\eta)})=\mathbb Z, $$
the complexification of the Lie algebra of the Hodge group $\operatorname{Hg}(M_\eta\otimes_{\kappa(\eta)}\mathbb {C})$ is a simple Lie algebra of type $C_{d-1}$, all bad reductions of the Abelian variety $\mathcal M_\eta$ are semi-stable, for any places $\delta,\delta'$ of bad reductions the $\mathbb Q$-space of Hodge cycles on the product $\operatorname{Alb}(\overline{\mathcal M_\delta^0})\,\times \, \operatorname{Alb}(\overline{\mathcal M_{\delta'}^0})$ of Albanese varieties is generated by classes of algebraic cycles, then there exists a finite ramified covering $\widetilde{C}\to C$ such that, for any Künnemann compactification $\widetilde{X}$ of the Néron minimal model of the Abelian variety $\mathcal M_\eta\otimes_{\kappa(\eta)}\kappa(\widetilde{\eta})$, the Grothendieck standard conjecture $B(\widetilde{X})$ of Lefschetz type is true.