For $a:a_1\geq \cdots\geq a_c$, let $\sigma_a$ be the corresponding Schubert cycle over $Gr(c,\infty)$. We say $a$ is of depth $k$ if $a_1-a_c=k$ ($c>1$). Let $a$ and $b$ be of depth $k_1$ and $k_2$, respectively. Is the following true?
Among all Schubert cycles inside the expansion of $\sigma_a\cdot \sigma_b$, there is one whose depth is at most $\max(k_1,k_2)$.
One need to use Littlewood-Richardson rule cleverly!