Working hypothesis on embedding partial orders into the Rudin–Keisler ordering of P-points

Let $\mathsf{MA}(\sigma\text{-centered})$ hold, and let $(P,<)$ be a partial order of size at most $2^{\mathfrak{c}}$ such that every $p\text{in}P$ has at most $\mathfrak{c}$ many predecessors. Working hypothesis. The partial order $P$ embeds into the Rudin–Keisler ordering of P-point ultrafilters. The statement is presented as a working hypothesis motivated by the known cardinality restrictions on the ordering of P-points; the text says that a stronger version asks for embeddings into both the Rudin–Keisler and Tukey orderings, and attributes this formulation in question form to Raghavan and Shelah.