Let \(\mathcal{M}\rightarrow C\) be the Néron minimal model of a 4-dimensional principally polarized abelian variety \(\mathcal{M}_{\eta}\) with trivial trace over the field \(k(\eta)\) of rational functions on a smooth projective curve \(C\). Suppose that \(\mathcal{M}_{\eta}\) is isomorphic to the product of absolutely simple abelian varieties over \(k(\eta)\) such that, for every simple factor \(I_{\overline{\eta}}\) of dimension greater than 2, the center of its endomorphism ring is not a CM-field nor a definite quaternion division algebra over \(\mathbf{Q}\). In this situation the author proves that there exists a finite ramified covering \(\tilde{C}\rightarrow C\) such that for any Kunnemann compactification \(\tilde{X}\) of the Néron minimal model of \(\mathcal{M}_{\eta}\otimes_{k(\eta)}k(\tilde{\eta})\) there are algebraic isomorphisms \(H^8(\tilde{X},\mathbb{Q})\cong H^2(\tilde{X},\mathbb{Q})\) and \(H^7(\tilde{X},\mathbb{Q})\cong H^3(\tilde{X},\mathbb{Q})\). Moreover assume that \(\mathcal{M}_{\overline{\eta}}\) has no complex multiplication, and that all bad reductions of \(\mathcal{M}_{\eta}\) are semi-stable with toric rank 1, and for any places \(\delta, \delta'\in C\) of bad reductions, the Hodge conjecture holds for the product \(A_{\delta}\times A_{\delta'}\) whose factors are the quotients of the connected components of neutral elements in special fibres of the Néron model modulo toric parts. Then he proves that the Grothendieck standard conjecture \(B(\tilde{X})\) of Lefschetz type holds true.