A conjecture about set partitions

I offer $1 for a proof of the following conjecture of mine, and $0.50 for a counter-example.

Let $\displaystyle S$ be a set with $\displaystyle mn$ elements. Let $\displaystyle P_1$ and $\displaystyle P_2$ be partitions of $\displaystyle S$ into $\displaystyle m$ parts, each having $\displaystyle n$ elements.

A subset of $\displaystyle S$ is called a "set of representatives" for a partition $\displaystyle P$ if each part of $\displaystyle P$ contains exactly one representative.

I conjecture that there is a subset of $\displaystyle m$ elements of $\displaystyle S$ which is a set of representatives both for $\displaystyle P_1$ and for $\displaystyle P_2$.