  For a family of subsets of $[n]:={1,2,...,n}$, the Lubell function is defined
as $\hb_n(\F):=\sum_{F\in\F}\binom{n}{|F|}^{-1}$. In \cite{GriLiLu}, Griggs,
Lu, and the author conjectured that if a family $\F$ of subset of $[n]$ does
not contain four distinct sets $A$, $B$, $C$ and $D$ forming a diamond, namely
$A\subset B\cap C$ and $B\cup C\subset D$, then $\hb_n(\F)\le
2+\lfloor\frac{n^2}{4}\rfloor/(n^2-n)$. Moreover, the upped bound is achieved
by three types of families.
  In this paper, we prove the upper bound in the conjecture is asymptotically
correct. In addition, we give some results related to the problem of maximizing
the Lubell function for the poset-free families.
