The problem Counting cycles after permuting within rows and columns reminds me of the
following unpublished conjecture of mine. Let $D$ be *any* finite
planar diagram (in the sense of Young diagram, which is a special
case), say with $n$ squares. Put the numbers $1,2,\dots,n$ into the
squares of the diagram. Let $R_D$ be the subgroup of the symmetric
group $S_n$ permuting elements within each row, and similarly $C_D$
for the columns. Let $\chi$ and $\psi$ be any characters of
$S_n$. Define
$$ u_D=\sum_{\substack{u\in R_D\\ v\in C_D}}
\chi(u)\psi(v)p_{\rho(uv)}, $$
where $p_{\rho(uv)}$ is the power sum symmetric function indexed by
the cycle type of $uv$. Then (conjecturally) $u_D$ is Schur-positive.

This conjecture is open even for diagrams of partitions when $\chi$ and $\psi$ are the trivial character. (In this case, one can show for hook shapes that $u_D$ is even $h$-positive, but $h$-positivity fails in most other cases.) When $D$ is the diagram of a partition $\lambda$, and where $\chi$ is the trivial character and $\psi$ the sign character, we have $u_D= H_\lambda s_\lambda$, where $H_\lambda$ is the product of the hook lengths of $\lambda$. See the slides of Valentin Féray at http://fpsac.combinatorics.kr/program.