  An old conjecture of Durfee 1978 bounds the ratio of two basic invariants of
complex isolated complete intersection surface singularities: the Milnor number
and the singularity (or geometric) genus. We give a counterexample for the case
of non-hypersurface complete intersections, and we formulate a weaker
conjecture valid in arbitrary dimension and codimension. This weaker bound is
asymptotically sharp.
  In this note we support the validity of the new proposed inequality by its
verification in certain (homogeneous) cases. In our subsequent paper we will
prove it for several other cases and we will provide a more comprehensive
discussion.
