Let $X$ be a set and ${\cal P}(X)$ its powerset. We say that ${\cal F} \subseteq {\cal P}(X)$ has the *splitting property (SP)* if there is $A\in {\cal P}(X)$ such that for all $F\in {\cal F}$ we have $$F \cap A \neq \emptyset \neq F\cap (X\setminus A).$$

Let $\text{SP}(X)$ denote the collection of all subsets of ${\cal P}(X)$ with (SP), and we order it with $\subseteq$.

If $X$ is infinite, and ${\cal F}\in \text{SP}(X)$, is there ${\cal M}\in\text{SP}(X)$ such that ${\cal M}$ is maximal in $(\text{SP}(X),\subseteq)$ and ${\cal F}\subseteq {\cal M}$?

(The obvious tool to try to use, Zorn's Lemma seems to be of no help in this, but I might be wrong.)

i.e., the hypergraph $(X,\mathcal F)$ has chromatic number $\le2.$ $\endgroup$