The EGH Conjecture and the Sperner property of complete intersections

Let $A$ be a graded complete intersection over a field and $B$ the monomial complete intersection with the generators of the same degrees as $A$. The EGH conjecture says that if $I$ is a graded ideal in $A$, then there should be an ideal $J$ in $B$ such that $B/J$ and $A/I$ have the same Hilbert function. We show that if the EGH conjecture is true, then it can be used to prove that every graded complete intersection over any field has the Sperner property.