  The Milnor number, \mu(X,0), and the singularity genus, p_g(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 \mu(X,0) \geq (n+1)!p_g(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.
