Erdős problem 623

Let $X$ be a set of cardinality $\aleph_\omega$ and $f$ a function from the finite subsets of $X$ to $X$ such that $f(A)\not\in A$ for all $A$. Must there exist an infinite independent $Y\subseteq X$, i.e. with $f(B)\not\in Y$ for all finite $B\subset Y$? Claimed resolution: the positive assertion is equivalent to Koepke's free-subset property, hence independent of ZFC, with consistency strength exactly a measurable cardinal.