Dicks–Ventura retract inertia conjecture for free groups

Let $F_n$ be a free group of finite rank $n\geq 2$. A subgroup $R\leq F_n$ is a retract of $F_n$ if there is a homomorphism $r:F_n\to R$ whose restriction to $R$ is the identity. A subgroup $H$ of $F_n$ is inert if $\operatorname{rk}(K\cap H)\leq \operatorname{rk}(K)$ for every subgroup $K$ of $F_n$.

Dicks–Ventura conjecture. Every retract of $F_n$ is inert.

Turner’s argument shows that this is equivalent to the Dicks–Ventura conjecture asserting inertia of fixed subgroups of arbitrary families of endomorphisms. The conjecture is known for fixed subgroups of families of injective endomorphisms, but remains open for arbitrary endomorphisms.