The Milnor number, μ(X, 0) , and the singularity genus, pg(X, 0) , are fundamental invariants of isolated hypersurface singularities (more generally, of local complete intersections). The long standing Durfee conjecture (and its generalization) predicted the inequality μ(X, 0) ≥ (n+ 1) ! pg(X, 0) , here n= dim (X, 0). Recently we have constructed counterexamples, proposed a corrected bound and verified it for the homogeneous complete intersections. In the current paper we treat the case of germs with Newton-non-degenerate principal part when the Newton diagrams are “large enough”, i.e. they are large multiples of some other diagrams. In the case of local complete intersections we prove the corrected inequality, while in the hypersurface case we prove an even stronger inequality.