The finite-character-subset conjecture for perfect phylogenies

Let $X$ be a finite set of taxa. An $r$-state character on $X$ is a partition of $X$ into at most $r$ states. A set $C$ of characters is compatible if there is a tree on $X$ that displays every character in $C$; such a tree is a perfect phylogeny for $C$.

Finite-character-subset conjecture. For each positive integer $r$, there exists an integer $f(r)$ such that, for any finite set $X$ and any set $C$ of $r$-state characters on $X$, $C$ is compatible if and only if every subset of at most $f(r)$ characters in $C$ is compatible.

This conjecture formalizes the question of whether compatibility can always be determined by checking character subsets of bounded size, independently of the total number of characters. The paper states that the conjecture is false, so it is refuted.