On the standard conjecture for projective compactifications of Néron models of $3$-dimensional Abelian varieties

We prove that the Grothendieck standard conjecture of Lefschetz type holds for a smooth complex projective $4$-dimensional variety $X$ fibred by Abelian varieties (possibly, with degeneracies) over a smooth projective curve if the endomorphism ring $\operatorname{End}_{\overline{\kappa(\eta)}} (X_\eta\otimes_{\kappa(\eta)}\overline{\kappa(\eta)})$ of the generic geometric fibre is not an order of an imaginary quadratic field. This condition holds automatically in the cases when the reduction of the generic scheme fibre $X_\eta$ at some place of the curve is semistable in the sense of Grothendieck and has odd toric rank or the generic geometric fibre is not a simple Abelian variety.