  We address the conjecture of [Durfee1978], bounding the singularity genus,
p_g, by a multiple of the Milnor number, \mu, for an n-dimensional isolated
complete intersection singularity. We show that the original conjecture of
Durfee, namely (n+1)!p_g\leq \mu, fails whenever the codimension r is greater
than one. Moreover, we propose a new inequality, and we verify it for
homogeneous complete intersections. In the homogeneous case the inequality is
guided by a `combinatorial inequality', that might have an independent
interest.
