  Given a graded poset $P$, consider a chain decomposition $\mathcal{C}$ of
$P$. If $|C_1|\le |C_2|$ implies that the set of the ranks of elements in $C_1$
is a subset of the ranks of elements in $C_2$ for any chains $C_1,C_2\in
\mathcal{C}$, then we say $\mathcal{C}$ is a nested chain decomposition (or
nesting, for short) of $P$, and $P$ is said to be nested. In 1970s, Griggs
conjectured that every normalized matching rank-unimodal poset is nested. This
conjecture is proved to be true only for all posets of rank 2 [W:05], some
posets of rank 3 [HLS:09,ENSST:11], and the very special cases for higher
ranks. For general cases, it is still widely open. In this paper, we provide
some sufficient conditions on the rank numbers of posets of rank 3 to satisfies
the Griggs's conjecuture.
