On the standard conjecture for compactifications of Néron models of 4-dimensional Abelian varieties

We prove that, after lifting to some finite ramified covering of a smooth projective curve $C$, the Grothendieck standard conjecture of Lefschetz type holds for the Künnemann compactification of the Néron minimal model of a 4-dimensional principally polarized Abelian variety over the field of rational functions on the curve $C$ provided that the endomorphism ring of the generic geometric fibre of the Néron model coincides with the ring of integers, all bad reductions are semi-stable and have toric rank 1 and, for any places $\delta,\delta'\in C$ of bad reductions, the Hodge conjecture on algebraic cycles holds for the product $A_\delta\times A_{\delta'}$ of the Abelian varieties $A_\delta,A_{\delta'}$ which are the quotients of the connected components of neutral elements in special fibres of the Néron minimal model modulo toric parts.