$\DeclareMathOperator{\NPU}{\operatorname{NPU}}\DeclareMathOperator{\RK}{\,\mathrm{RK}}$A *pre-ordered* set is a pair $(P, \leq)$ where $P$ is a set and $\leq\subseteq P\times P$ is a reflexive and transitive relation.

Let $\NPU(\omega)$ be the set of non-principal ultrafilters on $\omega$. The *Rudin-Keisler pre-order* on $\NPU(\omega)$ is defined by
$${\cal U} \leq_{\RK} {\cal V} :\Leftrightarrow (\exists f:\omega\to\omega)(\forall U\in{\cal U}) f^{-1}(U)\in {\cal V} .$$

It is easy to see that $\leq_{\RK}$ is reflexive and transitive, but not anti-symmetric.

**Question.** Given a finite pre-ordered set $(P,\leq)$, is there a finite subset $S\subseteq\NPU(\omega)$ such that $(P,\leq)$ is isomorphic to $S$ with the pre-order inherited from $\text{NPU}(\omega)$?