A question of B. Plotkin about the semigroup of endomorphisms of a free group

Let $F$ be a free group of finite rank $n \geq 2$, let $End(F)$ be the semigroup of endomorphisms of $F$, and let $Aut(F)$ be the group of automorphisms of $F$.

Theorem. If $T : End(F) \to End(F)$ is an automorphism of $End(F)$, then there is an $\alpha \in Aut(F)$ such that $T(\beta) = \alpha \circ \beta \circ \alpha^{-1}$ for all $\beta \in End(F)$.