Test-element conjecture for finitely generated subgroups of free groups

Let $F_n$ be a free group, and let $H$ be a finitely generated subgroup of $F_n$ that is not contained in a proper retract of $F_n$. A test element of a group is an element that is fixed only by endomorphisms acting as the identity on the whole group.

Test-element conjecture. Every test element of $H$ is a test element of $F_n$.

The source attributes this conjecture to an earlier work. It concerns how the retract-theoretic position of $H$ controls the endomorphism-detecting elements it contains; its resolution is not indicated in the supplied text.