Invariant-subsemigroup separation conjecture for free groups

Let $C$ be a non-empty finite set, let $C^+$ be the set of non-empty finite words over $C$, and let $F_C$ be the free group on $C$. An invariant sub-semigroup $S$ of $F_C$ is closed under multiplication and conjugation by elements of $F_C$.

Invariant-subsemigroup separation conjecture. For every $P\subseteq C^+$ such that $P$ and $C^+\setminus P$ are closed under concatenations and cyclic shifts, there exists an invariant sub-semigroup $S$ of $F_C$ such that $C^+\setminus P\subseteq S$, $P\cap S=\varnothing$, and, for every $g\in F_C$, either $g\in S$ or $g^{-1}\in S$.

The paper states this as a reduction of the ordered-group representation conjecture to a question about free groups. It is not proved in the supplied text.