The Saturation Conjecture (proved by Knutson-Tao) asserts that
$c_{n\mu,n\nu}^{n\lambda}\neq 0\Rightarrow c_{\mu,\nu}^{\lambda} \neq
0$, where $c$ denotes a Littlewood-Richardson coefficient and $n$ is a
positive integer. Let $\mathfrak{o}$ be a (commutative) discrete valuation
ring
with a finite residue field $k=\mathfrak{o}/\mathfrak{p}$. Let $M$ be
a finite $\mathfrak{o}$-module. (For instance, we can take
$\mathfrak{o}$ to be the $p$-adic integers $\mathbb{Z}_p$, in which
case $M$ is a finite abelian $p$-group.) Thus there is a unique partition
$\lambda=(\lambda_1,\lambda_2,\dots,\lambda_r)$ such that
$$ M \cong \bigoplus_{i=1}^r
(\mathfrak{o}/\mathfrak{p}^{\lambda_i}). $$
We call $\lambda$ the *type* of $M$. A fundamental result of Philip
Hall is that if $M$ has type $\lambda$, then there exists a submodule
$N$ of type $\mu$ and cotype $\nu$ (i.e., $M/N$ has type $\nu$) if and
only if $c^{\lambda}_{\mu\nu}\neq 0$. See Macdonald, *Symmetric
Functions and Hall Polynomials*, Chapter II, (4.3).

Now suppose that $M$ has type $n\lambda$, and $N$ has type $n\mu$ and cotype $n\nu$. Is there is a submodule $L$ of $M$ of type $\lambda$, such that $L\cap N$ has type $\mu$ and cotype (with respect to $L$) $\nu$?